research article tundish cover flux thickness measurement...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 690948 15 pageshttpdxdoiorg1011552013690948

Research ArticleTundish Cover Flux Thickness Measurement Methodand Instrumentation Based on Computer Vision in ContinuousCasting Tundish

Meng Lu He Qing Xie Zhi Yang Weimin Ci Ying Zhang Chuanyi and Gao Hongliang

College of Information Science and Engineering Northeastern University Shenyang 110000 China

Correspondence should be addressed to Meng Lu menglu1982gmailcom

Received 10 April 2013 Accepted 2 July 2013

Academic Editor Praveen Agarwal

Copyright copy 2013 Meng Lu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Thickness of tundish cover flux (TCF) plays an important role in continuous casting (CC) steelmaking process Traditionalmeasurement method of TCF thickness is singledouble wire methods which have several problems such as personal securityeasily affected by operators and poor repeatability To solve all these problems in this paper we specifically designed and builtan instrumentation and presented a novel method to measure the TCF thickness The instrumentation was composed of ameasurement bar amechanical device a high-definition industrial camera a Siemens S7-200 programmable logic controller (PLC)and a computer Our measurement method was based on the computer vision algorithms including image denoising methodmonocular rangemeasurementmethod scale invariant feature transform (SIFT) and image gray gradient detectionmethod Usingthe present instrumentation and method images in the CC tundish can be collected by camera and transferred to computer to doimaging processing Experiments showed that our instrumentation and method worked well at scene of steel plants can accuratelymeasure the thickness of TCF and overcome the disadvantages of traditional measurementmethods or even replace the traditionalones

1 Introduction

As the development of high-speed casting tundish cover flux(TCF) is becoming the key material for continuous castingprocess which has more and more important and irreplace-able roles on the quality of steel billet The main functionsof TCF are as follows isolate the steel surface to preventfreezing protect the steel surface from oxidation absorbinclusions that are transported to the surface maintain a lowfriction between steel shell andmold by lubrication create anoptimum heat transfer from steel to mold and so on [1ndash4]

The melting point of TCF (about 1000sim1150∘C) is lowerthan the temperature of molten steel (about 1500∘C) And theTCF is divided into threemain parts according to the physicalform namely liquid slag layer sintered layer and powder slaglayer (shown in Figure 1) TCF is added on the top of themolten steel in the continuous casting mold which partiallymelts and forms a liquid slag layer (about 6sim15mm) above themolten steel The liquid slag layer can partly prevent the heat

of molten steel from transferring therefore above the liquidslag layer temperature drops which forms a sintered layer(about 600sim900∘C) Above the sintered layer more heat isprevented therefore the TCF can keep its original pulverouscharacteristic to form a powder slag layer The powder slaglayer covers the molten steel surface evenly to prevent heatradiation and isolate oxygen

In the continuous casting steelmaking process due tocrystallizer vibration and solidified shell movement theliquid slag layer may slowly get into the gap between the shelland the copper wall of crystallizer Therefore along with thecasting process the liquid slag layer is constantly consumedby the molten steel and gets supplement from the sinteredlayer Similarly the sintered layer is constantly transformedinto liquid slag layer and gets supplement from the powderslag layer And the operator keeps an eye on the powderslag layer and adds new TCF into CC tundish as necessaryThe thickness of TCF has a great effect on the continuouscasting steelmaking process if the TCF is too thin the steel

2 Mathematical Problems in Engineering

CC tundish

Molten steel

Liquid slag layer

Sintered layer

Powder slag layer

Figure 1 Schematic diagram of TCF three main components in theCC tundish which are powder slag layer sintered layer and liquidslag layer

will become carburized unmelted slag patches will occur assurface defects and worse of all the result can be a breakoutbecause of too high friction between steel shell and mouldOn the contrary if the TCF is too thick crystallized rim willemerge at the meniscus which will disturb the solidificationprocess and cause surface defects on the steelTherefore TCFthickness is an essential parameter to guarantee the quality ofstrand surface and can provide important information for theoperators to keep track of the status in the CC tundish

Nowadays in continuous casting steelmaking processthe most common used measurement methods for TCFthickness are single-wire measurement method and double-wire measurement method because these two methodsare cheap and easily operated In single-wire measurementmethod one iron wire is vertically inserted into the moltensteel and kept steady for 3sim5 seconds Then pull it outand measure the TCF thickness based on the length ofcolor change and slag-adhering part (shown in Figure 2)In double-wire measurement method one iron wire andone copper wire which have the same length are paralleland firmly tied together then vertically inserted into themolten steel and kept steady for 3sim5 seconds Meltingpoint of the iron wire is higher than the temperature ofliquid slag layer and lower than the temperature of themolten steel therefore the iron wire below the liquid slaglayer is melted Melting point of the copper wire is higherthan the temperature of sintered layer and lower than thetemperature of liquid slag layer therefore the copper wirebelow the sintered layer is melted The whole thickness ofTCF can be calculated based on the length of the colorchange and slag-adhering part on the iron wire Comparedwith the singe-wire measurement method the double-wiremeasurement method can specifically measure the thicknessof the liquid slag layer using the length of the remaining ironwire minus the length of the remaining copper wire (shownin Figure 3) Although the single-wire measurement methodand the double-wire measurement method are widely usedin continuous casting steelmaking process they both haveinsurmountable and severe shortcomings (1) operators needto stand on the tundish to hold the wires which means

Iron wire

Color change and slage which adheres to the iron wire

Melted by molten steel

Metre rule

Figure 2 Single-wire measurement method of TCF thickness TCFthickness can be measured by the length of color change and slag-adhering part on the iron wire

Iron wire

Copper wire

Length difference of two wires

Metre rule

Figure 3 Double-wiremeasurementmethod of TCF thicknessThewhole thickness of the TCF can be measured by the length of thecolor change and slag-adhering part on the iron wire The thicknessof the liquid slag layer can be measured by the length difference oftwo wires

lethal danger to operators (2) the detection results are easilyinfluenced by the operators such as hand trembling or notvertically holding the wires (3) poor repeatability All theseshortcomings make these two TCF thickness measurementmethods inaccurate and dangerous

In this paper a novel TCF thickness measurementmethod and instrumentation are proposed We use a high-definition industrial camera to collect the image informa-tion in the CC tundish which is fastened to a custom-designed mechanical device The thickness of TCF is cal-culated based on the computer vision algorithms includ-ing image denoise method monocular range measurementmethod scale invariant feature transform (SIFT) and imagegray gradient detection method Our measurement methodis noncontact measurement full automatic working goodrepetitiveness and absolutely safe

Mathematical Problems in Engineering 3

10

12

3

4

5

6

7

89

Figure 4 Schematic diagram of the continuous temperature meas-urement systembased on the blackbody radiation theorem (1)Tem-perature sensor (2) optical lens (3) internal tube (4) external tube(5) molten steel (6) tundish (7) bracket (8) single-chip micyoco(SCM) (9) signal processor and (10) display for temperature

2 Method

The TCF thickness measurement method and instrumenta-tion are an extension of a previous work from our group [5ndash21] which is continuous temperature measurement methodand sensor formolten steel inCC tundish based on blackbodychamber theorem We briefly recall it in order to make thefollowing sections more clearly

Theoretical principle of continuous temperature mea-surement method is blackbody radiation theorem

119864

119887= (120582 119879

119871) = 120576

119879

(120582 119879) sdot 119864

119887(119879

0 120582) (1)

119864

119887represented the spectral radiosity of blackbody chamber

120576

119879 represented the spectral emissivity of blackbody chamber119879

119871represented the brightness temperature of blackbody

chamber and 119879 represented the actual temperature of black-body chamber

Based on this theorem we specifically made a measure-ment bar and used it as a temperature sensor which wasinserted into the molten steel to measure the temperature ofmolten steel The measurement bar was the core componentof the system which was composed of external tube andinternal tube The external tube was heat resistant shockresistant and anti corrosion and had good heat conductionwhich can offer protection for the internal tube The inter-nal tube was made of certain translucent medium whichhad steady radiation characteristic and speculardiffusionreflection characteristic Heat radiation in the measurementbar was transformed into electrical signal by a photoelectrictransducer And the electrical signal can be used to calculatethe temperature of the molten steel based on the blackbodyradiation theoremThe schematic diagram of the system wasshown in Figure 4

LLower

LUpper

Measurement bar

Mold powder

Molten steel

Camera optical center

Figure 5 Schematic diagram ofmeasurement bar which is insertedinto the molten steel

More details about the continuous temperature methodand sensor for molten steel in CC tundish can be referencedfrom [5ndash13]

21Theory Themeasurement bar is also the core componentof TCF thickness measurement method and instrumenta-tion While the measurement bar is inserted into the moltensteel a high-definition industrial camera is used to collect theimage information in the tundish Based on computer visionalgorithms which are elaborated in the Sections 23 and 24TCF thickness is calculated by

119871powder = 119871 lower minus 119871upper (2)

where 119871 lower represents the distance from TCF lower surfaceto optical center of the camera and 119871upper represents thedistance from TCF upper surface to optical center of thecamera (shown in Figure 5)

Molten steel and TCF have different thermal conductivi-ties therefore these two different media have different tem-peratures and there is obvious temperature gradient in theinterface layer of these two media While the measurementbar is inserted into the molten steel for long enough time(shown in Figure 5) the measurement bar can be in the stateof thermal balance And then the measurement bar is pulledup the temperature information of molten steel and TCF canbe reflected by the luminance of themeasurement bar (shownin Figure 6) In Figure 6(a) the brighter zone indicates thatthis part of the measurement bar is in the molten steel beforebeing pulled up and the darker zone indicates that this partof the measurement bar is in the TCF before being pulledup The arrow indicates the peak value of image gradientbetween the brighter zone and the darker zone which is alsothe interface layer between the molten steel and the TCFIn Figure 6(b) it can be seen that the temperature on themeasurement bar has an intense change which causes thepeak value of the temperature gradient indicated by the arrowThese two arrows in both Figures 6(a) and 6(b) actually pointto the same location of the measurement bar which is thelower surface of the TCF Therefore the distance from TCFlower surface to optical center of the camera can be calculatedby gray gradient detection in the image which is elaboratedin Section 24


In the TCFbefore being

pulled up

In the moltensteel before

being pulled up

(a)

Measurement bar coordinate (mm)Te

mpe

ratu

re (∘

C)0 50 100 150 200 250 300

200

400

600

800

1000

1200

1400

1600

02468101214161820222426

Temperature distribution curveTemperature gradient curve

Tem

pera

ture

gra

dien

t (∘ C

mm

)

(b)

Figure 6 The surface condition of the measurement bar when the measurement bar is pulled up from the molten steel (a) Luminanceinformation on the measurement bar on which obvious image gray scale difference can be seen (b) Schematic diagram of temperaturedistribution and gradient curve on the measurement bar Horizontal ordinate represents coordinates of the measurement bar left verticalordinate represents the temperature of the measurement bar and the right vertical ordinate represents the temperature gradient of themeasurement bar

22 Instrumentation Asmentioned before themeasurementbar is inserted into the molten steel therefore a specializedmechanical device is required to hold the measurementbar and the camera is also fixed on the mechanical device(shown in Figure 7) The mechanical device can ascend anddescend under the control of frequency conversion motorand the measurement bar can be lifted and lowered with themovement of the mechanical device The camera is fixed andwell-secured on the crossbeam of the mechanical device andin case of overheating the camera is cooled down by cool airthrough ventilating pipe With the help of this mechanicaldevice the height of the crossbeam the temperature of themeasurement bar and images collected by the camera aretransferred to the computer and the programmable logiccontroller (PLC Siemens S7-200) We can use all theseinformation tomake judgments and calculate TCF thicknessThe main functions of the computer are imaging processingand displaying the calculation result of TCF thickness Themain functions of the PLC are controlling the movement ofthemechanical device andmaking it accurate and steadyTheschematic diagram of the whole instrumentation system isshown in Figure 8

23 TCF Upper Surface Distance Measurement The distancefrom TCF upper surface to the optical center of camerais calculated based on the principle of monocular distancemeasurement [22 23] Two images 119868

1 1198682can be obtained

while the camera is moved from position 1198751to position 119875

2 In

Measurement bar

Camera

Pulley

Steel chain

Frequency conversion motorSleeves

Pillar

Figure 7 Schematic diagram ofmechanical device which holds thecamera and measurement bar

Mechanical device

Measurement barTCF

Molten steel

Camera

Computer

Siemens S7-200

Figure 8 Schematic diagram of the whole instrumentation systemincluding computer Siemens S7-200 mechanical device cameraand measurement bar


y

xP998400(px py)

H

W

ab

c d

g

Figure 9 Schematic diagram of image coordinate system 119892 is thebase point which is in the center of the image 1198751015840 represents onepixel in the image with coordinate (119901

119909

119901

119910

) x-g-y represents therectangular coordinate systemW represents the width of the imageand H represents the height of the image

these two images the same objects have different coordinatesdue to the camera movement According to the coordinatedifferences and the movement distance from 119875

1to 1198752 the

distance from the objects to the optical center of camera canbe calculated And there are two key steps in the calculation(1) the transformation from image coordinates to the worldcoordinates (2) matching the corresponding pixels of thesame objects in two images 119868

1and 1198682

231 Transformation from the Image Coordinates to theWorld Coordinates Suppose that the image coordinates aredefined as (119901

119909 119901

119910) and the world coordinates are defined

as (119875

119909 119875

119910 119875

119911) We can see that the image coordinates

are two-dimensional and the world coordinates are three-dimensional therefore one of the world coordinates shouldbe set as constant The base point of the image coordinatesystem is defined as the center of the image (shown inFigure 9) The base point of the world coordinate system isdefined as the intersection point of TCF upper surface andthe optical axis originated from the camerarsquos optical center(shown in Figure 10) therefore 119875

119911equiv 0 The transformation

from image coordinates (119901119909 119901

119910) to the world coordinates

(119875

119909 119875

119910 119875

119911) is defined as formula (3) and the details of proof

can be obtained in appendixIn Figure 10 plane ABU represents TCF upper surface

ABCD represents the camerarsquos field of view point119874 representsthe camerarsquos optical center line OG represents the camerarsquosoptical axis point 119866 represents the intersection point ofoptical axis and plane ABU point 119866 is also the base point oftheworld coordinate system point 119868 represents the projectionpoint of optical center 119874 on the plane ABU and the distancefrom point119874 to point 119868 is hX-G-Y represents the rectangularcoordinate system 2120572

0represents the camerarsquos vertical field

angle 21205730represents the camerarsquos horizontal field angle and

120574

0represents the camerarsquos pitch angle Namely ang119864119874119865 = 2120572

0

line OG equally divides ang119864119874119865 ang119870119874119869 = 21205730 lineOG equally

divides ang119870119874119869 ang119866119874119868 = 120574

0 Point 119875 is any point on the

TCF upper surface and its world coordinate is (119875119909 119875

119910) in

the rectangular coordinate systemX-G-Y FrompointP draw

Y

X

A

EQ

P

K

G

L

J

D

F

C

IU

O

120572120573

1205740

B

Figure 10 Schematic diagram of world coordinate system

perpendicular line on the axis119866119884 and the perpendicular footis pointQ therefore 119875

119910= 119866119876 Connect point 119875 and pointU

and line PU intersects axis GX at point L therefore 119875119909= 119866119871

Connect point 119871 and optical center O and the intersectionangle of line OL and optical axis is 120573 Connect point 119876 andoptical center O and the intersection angle of line OQ andoptical axis is 120572

120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times (1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

119875

119911= 0

(3)

H represents the height of the imageW represents the widthof the image h represents the vertical distance from camerarsquosoptical center to the TCF upper surface 2120572

0represents

the camerarsquos vertical field angle 21205730represents the camerarsquos

horizontal field angle and 1205740represents the camerarsquos pitch

angleBased on formula (3) the world coordinates of any point

on the TCF upper surface can be calculated and the onlyunknown parameter is h which is the distance from the TCFupper surface to the camerarsquos optical center By moving thecamera vertically from ℎ

1to ℎ2(ℎ

2gt ℎ

1) although ℎ

1and

ℎ

2are unknown Δℎ = ℎ

1minus ℎ

2can be obtained (shown in

Figure 11) Suppose that the same object on the TCF uppersurface has one world coordinate 119875

1(119886 119887) before movement


O2

O1

h1 h2

G1G2Q I

1205721

1205722

Figure 11 Schematic diagram of moving camera from ℎ

1

to ℎ2

While the camera is moved from ℎ

1

to ℎ2

then the camerarsquos opticalcenter is moved from 119874

1

to 1198742

after movement therefore the basepoint of world rectangular coordinate system is also moved from119866

1

to 1198662

and another world coordinate1198752(119888 119889) aftermovementWhile

the camera is moved from ℎ

1to ℎ2 then the camerarsquos optical

center is moved from 119874

1to 1198742after movement therefore

the base point of world rectangular coordinate system is alsomoved from 119866

1to 1198662 Because the movement of camera is

vertical two world rectangular coordinate systems have thesame 119910-axis and different 119909-axis therefore119875

1and1198752have the

same horizontal coordinates in the world coordinate systemnamely 119886 = 119888 Therefore the distance from TCF uppersurface to camera optical center can be calculated by formula(4)

ℎ

1

cos 1205740

times 119905119892120573

1

times (1 +

(119905119892 (120574

0+ 120572

1) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

=

ℎ

2

cos 1205740

times 119905119892120573

2

times(1+

(119905119892 (120574

0+ 120572

2) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

ℎ

2minus ℎ

1= Δℎ

120572

1= arc119905119892

2119901

1199101times 119905119892120572

0

119867

120573

1= arc119905119892

2119901

1199091

times 119905119892120573

0

119882

120572

2= arc119905119892

2119901

1199102

times 119905119892120572

0

119867

120573

2= arc119905119892

2119901

1199092

times 119905119892120573

0

119882

(4)

232 Matching the Corresponding Pixels of the Same Objectsin Two Images SIFT based image registration method isused to match the corresponding pixels of the same objects

in two images SIFT was published by Lowe in 1999 [24]and improved in 2004 [25] which was used to detect anddescribe local image features SIFT is an excellent featuredescriptor because it is invariant to uniform scaling orien-tation and partially invariant to affine distortion and illu-mination changes SIFTrsquos application includes object recog-nition robotic mapping and navigation image stitching 3Dmodeling gesture recognition video tracking individualidentification of wildlife and match moving

Scale space is a formal theory for handling image struc-tures at different scales fromphysical and biological vision byrepresenting an image as a one-parameter family of smoothedimages The main type of scale-space is the linear (Gaussian)scale-space which can be defined by

119871 (119909 119910 120590) = 119866 (119909 119910 120590) lowast 119868 (119909 119910) (5)

119868(119909 119910) represents one image lowast represents convolution and119866(119909 119910 120590) represents Gaussian filter function

119866 (119909 119910 120590) =

(12120587120590

2

) 119890

minus(119909

2+119910

2)

2120590

2

(6)

(119909 119910) represents image coordinates and 120590 represents scalelevelTherefore difference of Gaussian (DoG) scale space canbe defined as

119863(119909 119910 120590) = (119866 (119909 119910 119896120590) minus 119866 (119909 119910 120590)) lowast 119868 (119909 119910)

= 119871 (119909 119910 119896120590) minus 119871 (119909 119910 120590)

(7)

Once DoG images have been obtained keypoints areidentified as local minimamaxima of the DoG images acrossscales This is done by comparing each pixel in the DoGimages to its eight neighbors at the same scale and ninecorresponding neighboring pixels in each of the neighboringscales If the pixel value is the maximum orminimum amongall compared pixels it is selected as a candidate keypoint

Scale-space extreme detection produces too many key-point candidates some of which are unstable The next stepin the algorithm is to perform a detailed fit to the nearby datafor accurate location scale and ratio of principal curvaturesThis information allows points to be rejected that have lowcontrast (and are therefore sensitive to noise) or are poorlylocalized along an edge

The interpolation of keypoints is done using the quadrat-ic Taylor expansion of the DoG scale-space function

119863 (119883) = 119863 +

120597119863

119879

120597119883

119883 +

1

2

119883

119879120597

2

119863

120597119883

2

119883(8)

Then the location of the extreme 119883 is determined bytaking the derivative of this function with respect to 119883 andsetting it to zero

119883 = minus

120597

2

119863

minus1

120597119883

2

120597119863

120597119883

119863 (

119883) = 119863 +

1

2

120597119863

119879

120597119883

119883

(9)


TheDoG functionwill have strong responses along edgeseven if the candidate keypoint is not robust to small amountsof noise Therefore in order to increase stability the Hessianmatrix is used to eliminate the keypoints that have poorlydetermined locations but have high edge responses

119867 = [

119863

119909119909119863

119909119910

119863

119909119910119863

119910119910

]

Tr (119867) = 119863119909119909+ 119863

119910119910= 120572 + 120573

Det (119867) = 119863119909119909119863

119910119910minus (119863

119909119910)

2

= 120572120573

(10)

120572 represents bigger eigenvalue and 120573 represents smallereigenvalue Suppose that 120572 = 119903120573 We can get

119877 =

Tr (119867)2

Det (119867)=

(120572 + 120573)

2

120572120573

=

(119903120573 + 120573)

2

119903120573

2

=

(119903 + 1)

2

119903

(11)

It follows that for some threshold eigenvalue ratio 119903th if119877 for a candidate keypoint is larger than (119903th + 1)

2

119903th thatkeypoint is poorly localized and hence rejected

Each keypoint is assigned one or more orientations basedon local image gradient directions This is the key step inachieving invariance to rotation as the keypoint descriptorcan be represented relative to this orientation and thereforeachieve invariance to image rotation

119898(119909 119910)=radic(119871 (119909+1 119910)minus119871 (119909 minus 1 119910))

2

+(119871 (119909 119910 + 1)minus119871 (119909 119910 minus 1))

2

120579 (119909 119910) = tanminus1 (119871 (119909 119910 + 1) minus 119871 (119909 119910 minus 1)

119871 (119909 + 1 119910) minus 119871 (119909 minus 1 119910)

)

(12)

119898(119909 119910) represents the gradient magnitude and 120579(119909 119910) rep-resents the orientation

A set of orientation histograms are created on 4 times 4

pixel neighborhoods with 8 bins each These histograms arecomputed frommagnitude and orientation values of samplesin a 16 times 16 region around the keypoint such that eachhistogram contains samples from a 4 times 4 subregion of theoriginal neighborhood region The magnitudes are furtherweighted by a Gaussian function with 120590 equal to one half thewidth of the descriptor windowThe descriptor then becomesa vector of all the values of these histograms Since there are4 times 4 = 16 histograms each with 8 bins the vector has 128elements This vector is then normalized to unit length inorder to enhance invariance to affine changes in illumination

The SIFT feature descriptors extracted from two images119868

1 1198682while the camera is moved from position 119875

1to position

119875

2are shown in Figure 12

24 TCF Lower Surface Distance Measurement While themeasurement bar is pulled out due to the temperaturedifference there is temperature gradient on themeasurementbar And the temperature gradient is reflected as gray gradientin the image (shown in Figure 6) Therefore the distancefrom TCF lower surface to the camera optical center iscalculated based on gray gradient detection in the image

The measurement bar and the camera are both fixed inthemechanical device so they are in the relatively static stateTherefore in the image pixels in the middle line of mea-surement bar have constant distance to the camera opticalcenter Suppose that (119871

119909 119871

119910) represents pixel coordinate and

119889 represents the pixelrsquos distance to camera optical center Thefunction 119891 119871

119910rarr 119889 can be obtained (shown in Figure 13)

There is a pitch angle between camera optical axis andthe middle line of the measurement bar (215∘ in this paper)therefore pixels in the middle line of the measurement barhave different spatial resolutions (mmpixel) and the biggerthe vertical coordinate is the higher the spatial resolutionis Suppose that (119871

119909 119871

119910) represents pixel coordinate and sr

represents the pixelrsquos spatial resolution the function 119892

119871

119910rarr 119904119903 can be obtained (shown in Figure 14)In the continuous casting steelmaking process the range

of the TCF thickness is 5mmsim50mm Given the priorknowledge the functions 119891 and g and the distance from theTCF upper surface to the optical center the image regionof interest (ROI) on the measurement bar can be obtainedAnd in the ROI the Gaussian-Laplace operator is used todetect the gray gradient (shown in Figure 15) While the graygradient caused by TCF lower surface is detected distancefrom TCF lower surface to camera optical center can beobtained by using function f

25 Implementation The steps below outline the procedurefor measuring the thickness of TCF and the Steps (2)ndash(7) arecircularly implemented every 30 minutes

(1) Initialization Prepare the mechanical device move it tothe specified location check the status of the movement barand the camera and ensure that they are steadily fixed Thencontrol the mechanical device to insert the movement bar inthe molten steel

(2) Thermal Balance Keep the movement bar in the moltensteel long enough to make it in the thermal balance (30minutes in this paper) And save the current position as 119901

1

and current image as 1198681

(3) Pulling out the Measurement Bar Control the mechanicaldevice to pull out the measurement bar and save the currentposition as 119901

2and current image as 119868

2

(4) Distance Calculation from TCF Upper Surface to CameraOptical CenterUse formula (4) to calculate the distance fromTCF upper surface to camera optical center and save it as119871upper

(5) Distance Calculation from TCF Lower Surface to CameraOptical Center Use the Gaussian-Laplace operator to obtainthe image gray gradient caused by TCF lower surface and usethe function 119891 119871

119910rarr 119889 to calculate the distance fromTCF

lower surface to camera optical center and save it as 119871 lower

(6) TCF Thickness Calculation Use formula (2) to calculatethe TCF thickness


(a) (b)

(c)

Figure 12 Schematic diagram of matching the corresponding pixels in two images based on SIFT descriptors (a) image 1198681

in position 1198751

(b) image 119868

2

in position 1198752

after the measurement bar is pulled out and (c) matching the corresponding pixels in two images based on SIFTdescriptors

(7) End Measurement Control the mechanical device toinsert the measurement bar in the molten steel and thenimplement Step (2)

3 Experiments and Results

TheTCF thicknessmeasurement instrumentation has alread-y been installed and applied in four Chinese steel plantswhich are Nanjing Steel Plant located in Jiangsu provincesince 2010 Daye Steel Plant located in Hubei province since2011 Hanzhou Steel Plant located in Zhejiang province since2011 and Sanming Steel Plant located in Fujian province since2012The instrumentation works well in these four worksitesand Figure 16 shows the working condition at scene

31 Experiment of TCF Upper Surface Distance MeasurementWedid this experiment to verify the correctness and accuracyof the method discussed in Section 23 The experiment stepsare as follows

(1) We localized themechanical device in position1198751and

manually measured the distance from the camera tothe TCF upper surface (700mm in this experiment)and then saved the image in this position as 119868

1

(2) We lifted the mechanical device in position 1198752(1198752minus

119875

1= 50mm in this experiment) and then saved the

image in this position as 1198682

(3) We did image registration for images 1198681and 1198682based

on SIFT and matched the corresponding pixels in


0 100 200 300 400 500 600 700 800 900 1000

1500

1400

1300

1200

1100

1000

900

800

700

Ly

d

Figure 13 Schematic diagram of the relationship between pixelsrsquovertical coordinates in the middle line of the measurement bar andits distance to the camera optical center ∙ represents the samplingpoints

sr

0 100 200 300 400 500 600 700 800 900 1000

30

28

26

24

22

20

18

16

14

Ly

Figure 14 Schematic diagram of the relationship between pixelsrsquovertical coordinates in the middle line of the measurement bar andits spatial resolution ∙ represents the sampling points

these two images (226 pixels were matched in thisexperiment)

(4) We set the parameters in formula (4) Δℎ representslifted distance from 119875

1to 1198752(50mm in this experi-

ment) 21205720represents vertical angle of view (426∘ in

this experiment) 21205730represents horizontal angle of

view (50∘ in this experiment) 1205740represents angle of

pitch (215∘ in this experiment) and (1199011199091

119901

1199101

) and(119901

1199092 119901

1199102

) represent the coordinates of correspondingpixels in images 119868

1and 1198682 Table 1 showed the horizon-

tal and vertical angles of view of several kinds lenswhich had different sizes and focuses

Table 1 Horizontal and vertical angles of view of several kinds lenswhich have different sizes and focuses

Focus(mm)

Horizontalangle ofview for13 inch

Verticalangle ofview for13 inch



25mm 964∘ 862∘ 819∘ 729∘

28mm 899∘ 798∘ 756∘ 668∘

36mm 757∘ 66∘ 622∘ 543∘

4mm 699∘ 607∘ 57∘ 495∘

6mm 50∘ 426∘ 398∘ 342∘

8mm 385∘ 326∘ 304∘ 26∘

12mm 262∘ 221∘ 205∘ 175∘

16mm 198∘ 166∘ 154∘ 132∘

30mm 106∘ 89∘ 83∘ 7∘

60mm 53∘ 45∘ 41∘ 35∘

100mm 32∘ 27∘ 25∘ 21∘

200mm 16∘ 13∘ 12∘ 11∘

(5) We used formula (4) to calculate the distance fromcamera to TCF upper surface and compared thecalculated results with the actual distance (700mmin this experiment) The calculation and comparisondetails were shown in Table 2

From the comparison shown in Table 2 average deviationis 12mm and the biggest deviation is 22mm thereforewe can conclude that the method in Section 23 based onmonocular vision and image registration can accuratelycalculate the distance from the camera to TCF upper surface

32 Comparison betweenOurMethod and SingleDoubleWireMethods on TCFThicknessMeasurement In this experimentthe results of our method were compared with the traditionalmethods which are single-wire measurement method anddouble-wire measurement method Although singledoublewire measurement methods have several disadvantages suchas easily affected by the operator dangerous to the operatorand poor repeatability they are still the most common usedand the most accredited methods in continuous castingsteelmaking process Therefore we took these two methodsas the golden standard to verify the correctness and accuracyof our method

To improve the accuracy of the singledouble wire mea-surement methods and reduce the human interference fromthe operators as far as possible two operators were asked tojoin the experiment and strictly obey the following rules (1)standing on the secure spot and staying alert (2) verticallyinserting the wires into themolten steel (3) keeping the handsteady while the wires are in the molten steel (4) keepingthe wires in the molten steel for five seconds (5) pulling thewires out of the molten steel as quickly as possible while thetime is up One operator measured the TCF thickness usingone-wire measurement method and the other one measuredthe TCF thickness using two-wire measurement methodWhile our instrumentation measured the TCF thickness


(a) (b)

Figure 15 (a) Original image (b) result of gray gradient detection Arrows represent the gray gradient caused by TCF lower surface

(a) (b)

Figure 16 Working condition of our instrumentation at scene (a) in the idle condition (b) in the working condition

both operators simultaneously measured the TCF thicknessusing singledouble wire measurement methods In this waywe can obtain three groups of experimental data at the sametime to make the comparison one from our method twofrom singledouble wire measurement methods

The main ingredient of the TCF was shown in Table 3The operator was asked to add one bag of TCF through eachtundish hole every two hours and the weight of one bag ofTCF was about 10 kg

We did this experiment in two steel plants respectivelywhich are Sanming Steel Plant and Nanjing Steel PlantThe error range that can be accepted by the steel plants isplusmn5mm Because the variation of TCF thickness is slow wemeasured the TCF thickness every 30minutes using both ourmethod and the traditional singledouble wire measurement

methods The experiments took ten hours in Sanming SteelPlant and nine hours in Nanjing Steel Plant The measure-ment results and comparison between our method and thetraditional singledouble wire measurement methods wereshown in Table 4 (Sanming Steel Plant) and Table 5 (NanjingSteel Plant) and the comparisons were also shown in Figures17 and 18

It is shown in Table 4 that in the experiment of SanmingSteel Plant the biggest deviation of our method from single-wire measurementmethod is 57mm and themean deviationof our method from single-wire measurement method is30mm and in the experiment of Sanming Steel Plant thebiggest deviation of our method from double-wire mea-surement method is 63mm and the mean deviation of ourmethod from double-wire measurement method is 28mm


Table 2 Calculated distance from TCF upper surface to cameraoptical center and the comparison between calculated distance andactual distance which is manually measured

Imagecoordinatesbeforemovement

Imagecoordinates

aftermovement

Calculateddistance(mm)

Actualdistance(mm)

Deviation(mm)

(455 220) (450 345) 6986 700 minus14(492 224) (486 348) 7013 700 13(501 227) (494 351) 7015 700 15(520 251) (513 374) 7011 700 11(526 247) (518 370) 6985 700 minus15(534 252) (526 374) 7008 700 08(547 249) (539 372) 7000 700 0(551 248) (542 371) 7018 700 18(330 277) (335 402) 7009 700 09(474 239) (468 363) 6991 700 minus09(501 231) (495 355) 7013 700 13(513 240) (505 362) 6990 700 minus1(520 246) (513 369) 7014 700 14(528 254) (521 376) 7018 700 18(529 260) (522 382) 7022 700 22(535 255) (527 378) 7011 700 11(535 259) (527 382) 7010 700 1(540 257) (532 380) 7011 700 11(513 240) (505 362) 6984 700 minus16(501 225) (494 349) 7003 700 03

It is shown in Table 5 that in the experiment of NanjingSteel Plant the biggest deviation of our method from single-wiremeasurementmethod is 64mm and themean deviationof our method from single-wire measurement method is21mm and in the experiment of Nanjing Steel Plant thebiggest deviation of our method from double-wire mea-surement method is 76mm and the mean deviation of ourmethod from double-wire measurement method is 30mm

From Tables 4 and 5 we can conclude that our instru-mentation andmeasurementmethod can accurately measurethe TCF thickness and our method has higher measure-ment precision and even can replace singledouble wiremeasurement methods in continuous casting steelmakingprocess

It is shown in Figures 17 and 18 that these three groupsof measured results (our method single-wire measurementmethod and double-wire measurement method) have simi-lar trend As time went by the TCF was melted by the moltensteel on and on therefore the TCF thickness became thinnerand thinner until the operator added one more bag of TCFthrough the tundish holeThe operator added TCF every twohours (at 2 4 6 and 8 hours in Figures 17 and 18) thereforethe TCF thickness became much thicker after the TCF wasadded

TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)

Figure 17 Measurement results and comparison between ourmethod and the traditional singledouble wire measurement meth-ods All these data were obtained from Sanming Steel Plant ◻represents the results of our method ∙ represents the results ofsingle-wire measurement method and represents the results ofdouble-wire measurement method

4 Conclusion

This paper presents a novel measurement method and in-strumentation to measure the thickness of TCF in con-tinuous casting steelmaking process The method is basedon computer vision algorithms and the instrumentation isspecifically designed and built which was composed of ameasurement bar a mechanical device a high-definitionindustrial camera a PLC and a computer Experiments andresults showed that our measurement method and instru-mentation can accurately measure the thickness of TCF incontinuous casting steelmaking process and had good safetyfor operators Furthermore our measurement method andinstrumentation can replace traditional singledouble wiremeasurement methods

Appendix

Theproof of transformation from image coordinates toworldcoordinates

∵ Point 119868 is the joint point of optical

center 119874 on the plane 119860119861119880

there4 Line 119874119868 perp plane 119860119861119880

there4 Plane 119874119866119868 perp plane 119860119861119880

∵ Line 119869119866 perp line 119864119865


(A1)

Suppose that point 119875 is any point on the plane ABCD whoseworld coordinate is (119875

119909 119875

119910) And the corresponding point of


Table 3 Main ingredient of the TCF

Ingredient SiO2 MnO Al2O3 CaO MgO FeO Fe2O3 P2O5

Content () 117ndash15 31ndash4 1ndash16 49ndash53 31ndash10 11ndash17 69ndash8 17ndash2

Table 4 Measurement results and comparison between our method and the traditional singledouble wire measurement methods All thesedata were obtained from Sanming Steel Plant

TCF thicknessmeasured by ourmethod (mm)

TCF thickness measuredby single-wire

measurement methodfrom operator no 1 (mm)

TCF thickness measuredby double-wire


Deviation of our methodfrom single-wire

measurement method (mm)

Deviation of our methodfrom double-wire


261 239 248 22 13248 233 235 15 13207 188 221 19 minus14305 261 276 44 29291 246 276 45 15249 228 261 21 minus12228 218 241 1 minus13298 251 283 47 15286 229 273 57 13257 225 261 32 minus04205 236 253 minus31 minus48329 288 308 41 21298 253 262 45 36243 218 275 25 minus32209 185 265 24 minus56208 239 258 minus31 minus5351 318 288 33 63316 334 269 minus18 47291 284 266 07 25210 244 258 minus34 minus48Biggest deviation 57 63Mean deviation 30 28

119875 in the image is 1198751015840 whose image coordinate is (119901119909 119901

119910) As

shown in Figure 7 point119875 and point119876 have the same verticalcoordinate which isGQTherefore corresponding point of119876in the image is1198761015840 and11987610158401198661015840 = 119901

119910 which shown in Figure 19

119905119892120572 =

119866

1015840

119876

1015840

119874119866

1015840

=

119901

119910

119866

1015840

119864

1015840

119905119892 (ang119866

1015840

119874119864

1015840

)

=

119901

119910

(12)119867119905119892120572

0

=

2119901

119910times 119905119892120572

0

119867

(A2)

there4 120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

(A3)

Similarly

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A4)

119867 represents the height of the image

119874119866 =

ℎ

cos 1205740

∵ 119868119866 = ℎ times 119905119892120574

0

119868119876 = ℎ times 119905119892 (120574

0+ 120572)

there4 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

(A5)

there4 119875

119910= 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0) (A6)


Table 5 Measurement results and comparison between our method and the traditional singledouble wire measurement methods All thesedata were obtained from the Nanjing Steel Plant










335 358 336 minus23 minus01310 319 307 minus09 03300 276 298 24 02376 403 361 minus27 15357 395 331 minus38 26312 300 285 12 27265 291 238 minus26 27400 409 376 minus09 24379 315 348 64 31310 276 297 34 13216 259 281 minus43 minus65438 429 381 09 57385 387 346 minus02 39289 316 308 minus27 minus19253 243 285 1 minus32455 458 425 minus03 3406 426 330 minus2 76371 371 321 0 5Biggest deviation 64 76Mean deviation 21 30

TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

464442

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)

Figure 18 Measurement results and comparison between ourmethod and the traditional singledouble wire measurement meth-ods All these data were obtained from Nanjing Steel Plant ◻represents the results of our method ∙ represents the results ofsingle-wire measurement method and represents the results ofdouble-wire measurement method

F998400

G998400

Q998400

E998400

O

h

IFGQEY

120572 1205740

Figure 19 Schematic diagram of the corresponding point verticalcoordinate between the world coordinates system and the imagecoordinates system Point 119876 has the same vertical coordinate withthe point119875 in the planeABCD and the corresponding point of point119876 in the image is point 1198761015840 and 11987610158401198661015840 = 119901

119910

From formula (A6) the vertical coordinate of point 119875 inthe world coordinates system can be obtained from the imagecoordinate system


∵ 119868119866 = ℎ times 119905119892120574

0

119868119865 = ℎ times 119905119892 (120574

0minus 120572

0)

there4 119866119865 = 119868119866 minus 119868119865 = ℎ times (119905119892120574

0minus 119905119892 (120574

0minus 120572

0))

∵ 119874119866 =

ℎ

cos 1205740

119874119865 =

ℎ

cos (1205740minus 120572

0)

and 119865119862119866119869 119869119866 perp plane 119874119866119868

there4 119865119862 perp plane 119874119866119868

there4 119866119869 = 119874119866 times 119905119892120573

0=

ℎ

cos 1205740

times 119905119892120573

0

119865119862 = 119874119865 times 119905119892120573

0=

ℎ

cos (1205740minus 120572

0)

times 119905119892120573

0

∵ Δ119880119865119862 ∽ Δ119880119866119869

there4

119865119862

119866119869

=

119880119865

119880119866

there4

119865119862 minus 119866119869

119866119869

=

119880119865 minus 119880119866

119880119866

=

minus119866119865

119880119866

(A7)

there4 119880119866 =

119866119869 times 119866119865

119866119869 minus 119865119862

=

ℎ [119905119892120574

0minus 119905119892 (120574

0minus 120572

0)] cos (120574

0minus 120572

0)

cos (1205740minus 120572

0) minus cos 120574

0

(A8)

there4 119866119871 = 119874119866 times 119905119892120573 =

119874119868

cos 1205740

times 119905119892120573 =

ℎ

cos 1205740

times 119905119892120573 (A9)

InΔ119880119875119876 119866119871119875119876

=

119880119866

119880119876

there4 119875119876 =

119866119871 (119880119866 + 119866119876)

119880119866

(A10)

GQ UG and GL can be obtained by formulas (A6) (A8)and (A9) respectively

Thus

there4 119875

119909= 119875119876

=

ℎ

cos 1205740

times 119905119892120573

times(1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

(A11)

From formula (A11) the horizontal coordinate of point119875 in the world coordinates system can be obtained from theimage coordinate system

Therefore the transformation from the image coordinatessystem (119901

119909 119901119910) to the world coordinates system (119875

119909 119875119910) is

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times(1 +

(119905119892 (120574

0+ 120572) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A12)

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (no 61101057) and Shenyang TaiheMetallurgical Measurement and Control Ltd

References

[1] H Kania K Nowacki and T Lis ldquoImpact of the density of themould powder on thickness of the layer of liquid slag in thecontinuous caster mouldrdquo Metalurgija vol 52 no 2 pp 204ndash206 2013

[2] H Kania and J Gawor ldquoImpact of mould powder density onsurface quality and near-surface zone microstructure of castslabrdquoArchives ofMetallurgy andMaterials vol 57 no 1 pp 339ndash345 2012

[3] J P BiratM Larrecq and J Y Lamant ldquoThe continuous-castingmoldmdasha basic tool for surface quality and strand productivityrdquoin Proceedings of the 74th Steelmaking Division of the Iron andSteel Society pp 39ndash50 1991

[4] K-D Schmidt F Friedel K-P ImlauW Jager andK TMullerldquoConsequent improvement of surface quality by systematicanalysis of slabsrdquo Steel Research International vol 74 no 11-12pp 659ndash666 2003

[5] Z Hu Y Ci and Z Xie ldquoMolten steel level measurement intundish with heat transfer analysisrdquo ISIJ International vol 51no 10 pp 1674ndash1681 2011

[6] Y Ci Z Xie and H Zhang ldquoNew approach to continuous tem-perature measurement of liquid steel in CC tundishrdquo Journal ofNortheastern University vol 25 no 5 pp 460ndash462 2004

[7] H Zhang and Z Xie ldquoAccuracy improvement and structureoptimization of continuous temperature measurement sensorfor liquid steelrdquo Acta Metrologica Sinica vol 29 no 1 pp 38ndash41 2008

[8] Z Xie Y Ci H Meng and H Zhang ldquoDevelopment ofcontinuous temperaturemeasuring sensor for liquid steel basedon blackbody cavityrdquo Chinese Journal of Scientific Instrumentvol 26 no 5 pp 446ndash449 2005

[9] H Zhang and Z Xie ldquoSimulation study on dynamic tem-perature measurement of liquid steel continuous temperaturemeasurement sensorrdquo Chinese Journal of Scientific Instrumentvol 28 no 10 pp 1775ndash1780 2007


[10] ZHua C Ying andX Zhi ldquoDynamicmodeling of temperaturesensors with great inertiardquo Chinese Journal of Scientific Instru-ment vol 25 no 4 pp 261ndash264 2004

[11] Z Shumao M Guohui Z Jiu and X Zhi ldquoFinite element anal-ysis of composite structure continuous temperature-measuringsensor for liquid steelrdquo Journal of Northeastern University(Natural Science) vol 33 no 7 pp 926ndash929 2012

[12] H Zhang and Z Xie ldquoStatic and dynamic uncertainty evalua-tion of continuous temperature measurement system for liquidsteelrdquo Acta Metrologica Sinica vol 28 no 4 pp 329ndash332 2007

[13] H-J Meng R-Y Wu Z Xie and G-Q Lin ldquoCasting speedoptimization control system based on continuous temperaturemeasurement of molten steel in CC tundishrdquo Journal of North-eastern University (Natural Science) vol 26 no 10 pp 942ndash9452005

[14] S-M Zhao G-H Mei and Z Xie ldquoEffect of MgO criticalparticle size on properties of MgO-C continuous temperature-measuring sensor for liquid steelrdquo Journal of Iron and SteelResearch International vol 18 no 12 pp 12ndash21 2011

[15] W Liu and Z Xie ldquoDesign and application of dynamic controlsystem for secondary cooling of billet continuous castingrdquo inProceedings of the IEEE International Conference on AdvancedComputer Control (ICACC rsquo10) pp 230ndash233 March 2010

[16] B Wang Z-P Ji W-H Liu J-C Ma and Z Xie ldquoApplicationof hot strength and ductility test to optimization of secondarycooling system in billet continuous casting processrdquo Journal ofIron and Steel Research International vol 15 no 4 pp 16ndash202008

[17] W-H Liu Z Xie Z-P Ji B Wang Z-Y Lai and G-L JialdquoDynamic water modeling and application of billet continuouscastingrdquo Journal of Iron and Steel Research International vol 15no 2 pp 14ndash17 2008

[18] J Ma Z Xie and G Jia ldquoApplying of real-time heat transferand solidificationmodel on the dynamic control system of billetcontinuous castingrdquo ISIJ International vol 48 no 12 pp 1722ndash1727 2008

[19] Z Ji and Z Xie ldquoMulti-objective optimization of continu-ous casting billet based on ant colony system algorithmrdquo inProceedings of the Pacific-Asia Workshop on ComputationalIntelligence and Industrial Application (PACIIA rsquo08) pp 262ndash266 December 2008

[20] W Liu Z Xie Z Ji and B Wang ldquoResearch and applicationof dynamic control system for secondary cooling of billetcontinuous castingrdquo in Proceedings of the 2nd IEEE Conferenceon Industrial Electronics and Applications (ICIEA rsquo07) pp 184ndash187 May 2007

[21] J Liu D Li Z Hu and Z Xie ldquoMonocular computer visionimage calibration method and its applicationrdquo in Proceedingsof the IEEE International Conference on Advanced ComputerControl (ICACC rsquo10) pp 144ndash147 March 2010

[22] G Lei X Youchun L Keqiang and L Xiaomin ldquoStudy on real-time distance detection based on monocular vision techniquerdquoJournal of Image and Graphics vol 11 no 1 pp 74ndash81 2006

[23] S W Yang S A Scherer and A Zell ldquoAn onboard monocularvision system for autonomous takeoff hovering and landing ofa micro aerial vehiclerdquo Journal of Intelligent amp Robotic Systemsvol 69 no 1ndash4 pp 499ndash515 2013

[24] D G Lowe ldquoObject recognition from local scale-invariantfeaturesrdquo inProceedings of the 7th IEEE International Conferenceon Computer Vision (ICCV rsquo99) pp 1150ndash1157 September 1999

[25] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60 no2 pp 91ndash110 2004

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


CC tundish

Molten steel

Liquid slag layer

Sintered layer

Powder slag layer

Figure 1 Schematic diagram of TCF three main components in theCC tundish which are powder slag layer sintered layer and liquidslag layer

will become carburized unmelted slag patches will occur assurface defects and worse of all the result can be a breakoutbecause of too high friction between steel shell and mouldOn the contrary if the TCF is too thick crystallized rim willemerge at the meniscus which will disturb the solidificationprocess and cause surface defects on the steelTherefore TCFthickness is an essential parameter to guarantee the quality ofstrand surface and can provide important information for theoperators to keep track of the status in the CC tundish

Nowadays in continuous casting steelmaking processthe most common used measurement methods for TCFthickness are single-wire measurement method and double-wire measurement method because these two methodsare cheap and easily operated In single-wire measurementmethod one iron wire is vertically inserted into the moltensteel and kept steady for 3sim5 seconds Then pull it outand measure the TCF thickness based on the length ofcolor change and slag-adhering part (shown in Figure 2)In double-wire measurement method one iron wire andone copper wire which have the same length are paralleland firmly tied together then vertically inserted into themolten steel and kept steady for 3sim5 seconds Meltingpoint of the iron wire is higher than the temperature ofliquid slag layer and lower than the temperature of themolten steel therefore the iron wire below the liquid slaglayer is melted Melting point of the copper wire is higherthan the temperature of sintered layer and lower than thetemperature of liquid slag layer therefore the copper wirebelow the sintered layer is melted The whole thickness ofTCF can be calculated based on the length of the colorchange and slag-adhering part on the iron wire Comparedwith the singe-wire measurement method the double-wiremeasurement method can specifically measure the thicknessof the liquid slag layer using the length of the remaining ironwire minus the length of the remaining copper wire (shownin Figure 3) Although the single-wire measurement methodand the double-wire measurement method are widely usedin continuous casting steelmaking process they both haveinsurmountable and severe shortcomings (1) operators needto stand on the tundish to hold the wires which means

Iron wire

Color change and slage which adheres to the iron wire

Melted by molten steel

Metre rule

Figure 2 Single-wire measurement method of TCF thickness TCFthickness can be measured by the length of color change and slag-adhering part on the iron wire

Iron wire

Copper wire

Length difference of two wires

Metre rule

Figure 3 Double-wiremeasurementmethod of TCF thicknessThewhole thickness of the TCF can be measured by the length of thecolor change and slag-adhering part on the iron wire The thicknessof the liquid slag layer can be measured by the length difference oftwo wires

lethal danger to operators (2) the detection results are easilyinfluenced by the operators such as hand trembling or notvertically holding the wires (3) poor repeatability All theseshortcomings make these two TCF thickness measurementmethods inaccurate and dangerous

In this paper a novel TCF thickness measurementmethod and instrumentation are proposed We use a high-definition industrial camera to collect the image informa-tion in the CC tundish which is fastened to a custom-designed mechanical device The thickness of TCF is cal-culated based on the computer vision algorithms includ-ing image denoise method monocular range measurementmethod scale invariant feature transform (SIFT) and imagegray gradient detection method Our measurement methodis noncontact measurement full automatic working goodrepetitiveness and absolutely safe


10

12

3

4

5

6

7

89


2 Method



119864

119887= (120582 119879

119871) = 120576

119879

(120582 119879) sdot 119864

119887(119879

0 120582) (1)

119864


120576





LLower

LUpper

Measurement bar

Mold powder

Molten steel










pulled up


being pulled up

(a)


mpe

ratu

re (∘

C)0 50 100 150 200 250 300

200

400

600

800

1000

1200

1400

1600

02468101214161820222426


Tem

pera

ture

gra

dien

t (∘ C

mm

)

(b)






2 In

Measurement bar

Camera

Pulley

Steel chain


Pillar


Mechanical device

Measurement barTCF

Molten steel

Camera

Computer

Siemens S7-200



y

xP998400(px py)

H

W

ab

c d

g


119909

119901

119910



1to 1198752 the


1and 1198682


119909 119901


as (119875

119909 119875

119910 119875






(119875

119909 119875

119910 119875






120574


0


divides ang119870119874119869 ang119866119874119868 = 120574



119910) in


Y

X

A

EQ

P

K

G

L

J

D

F

C

IU

O

120572120573

1205740

B






120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times (1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

119875

119911= 0

(3)


0represents





1to ℎ2(ℎ

2gt ℎ

1) although ℎ

1and

ℎ


1minus ℎ





O2

O1

h1 h2

G1G2Q I

1205721

1205722


1

to ℎ2


1

to ℎ2


1

to 1198742


1

to 1198662









1and1198752have the


ℎ

1

cos 1205740

times 119905119892120573

1

times (1 +

(119905119892 (120574

0+ 120572

1) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

=

ℎ

2

cos 1205740

times 119905119892120573

2

times(1+

(119905119892 (120574

0+ 120572

2) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

ℎ

2minus ℎ

1= Δℎ

120572

1= arc119905119892

2119901

1199101times 119905119892120572

0

119867

120573

1= arc119905119892

2119901

1199091

times 119905119892120573

0

119882

120572

2= arc119905119892

2119901

1199102

times 119905119892120572

0

119867

120573

2= arc119905119892

2119901

1199092

times 119905119892120573

0

119882

(4)




119871 (119909 119910 120590) = 119866 (119909 119910 120590) lowast 119868 (119909 119910) (5)


119866 (119909 119910 120590) =

(12120587120590

2

) 119890

minus(119909

2+119910

2)

2120590

2

(6)


119863(119909 119910 120590) = (119866 (119909 119910 119896120590) minus 119866 (119909 119910 120590)) lowast 119868 (119909 119910)

= 119871 (119909 119910 119896120590) minus 119871 (119909 119910 120590)

(7)




119863 (119883) = 119863 +

120597119863

119879

120597119883

119883 +

1

2

119883

119879120597

2

119863

120597119883

2

119883(8)


119883 = minus

120597

2

119863

minus1

120597119883

2

120597119863

120597119883

119863 (

119883) = 119863 +

1

2

120597119863

119879

120597119883

119883

(9)



119867 = [

119863

119909119909119863

119909119910

119863

119909119910119863

119910119910

]

Tr (119867) = 119863119909119909+ 119863

119910119910= 120572 + 120573

Det (119867) = 119863119909119909119863

119910119910minus (119863

119909119910)

2

= 120572120573

(10)


119877 =

Tr (119867)2

Det (119867)=

(120572 + 120573)

2

120572120573

=

(119903120573 + 120573)

2

119903120573

2

=

(119903 + 1)

2

119903

(11)


2




2

+(119871 (119909 119910 + 1)minus119871 (119909 119910 minus 1))

2


119871 (119909 + 1 119910) minus 119871 (119909 minus 1 119910)

)

(12)






1to position

119875




119909 119871





119909 119871



119871






1




2







(a) (b)

(c)


in position 1198751

(b) image 119868

2

in position 1198752








1


119875






0 100 200 300 400 500 600 700 800 900 1000

1500

1400

1300

1200

1100

1000

900

800

700

Ly

d


sr

0 100 200 300 400 500 600 700 800 900 1000

30

28

26

24

22

20

18

16

14

Ly









119901

1199101

) and(119901

1199092 119901

1199102





Focus(mm)





25mm 964∘ 862∘ 819∘ 729∘

28mm 899∘ 798∘ 756∘ 668∘

36mm 757∘ 66∘ 622∘ 543∘

4mm 699∘ 607∘ 57∘ 495∘

6mm 50∘ 426∘ 398∘ 342∘

8mm 385∘ 326∘ 304∘ 26∘

12mm 262∘ 221∘ 205∘ 175∘

16mm 198∘ 166∘ 154∘ 132∘

30mm 106∘ 89∘ 83∘ 7∘

60mm 53∘ 45∘ 41∘ 35∘

100mm 32∘ 27∘ 25∘ 21∘

200mm 16∘ 13∘ 12∘ 11∘






(a) (b)


(a) (b)










Imagecoordinates

aftermovement


Actualdistance(mm)

Deviation(mm)





TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


4 Conclusion


Appendix






∵ Line 119869119866 perp line 119864119865


(A1)


119909 119875


















119910) As



119905119892120572 =

119866

1015840

119876

1015840

119874119866

1015840

=

119901

119910

119866

1015840

119864

1015840

119905119892 (ang119866

1015840

119874119864

1015840

)

=

119901

119910

(12)119867119905119892120572

0

=

2119901

119910times 119905119892120572

0

119867

(A2)

there4 120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

(A3)

Similarly

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A4)


119874119866 =

ℎ

cos 1205740

∵ 119868119866 = ℎ times 119905119892120574

0

119868119876 = ℎ times 119905119892 (120574

0+ 120572)

there4 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

(A5)

there4 119875

119910= 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0) (A6)













TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

464442

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


F998400

G998400

Q998400

E998400

O

h

IFGQEY

120572 1205740


119910



∵ 119868119866 = ℎ times 119905119892120574

0

119868119865 = ℎ times 119905119892 (120574

0minus 120572

0)


0minus 119905119892 (120574

0minus 120572

0))

∵ 119874119866 =

ℎ

cos 1205740

119874119865 =

ℎ

cos (1205740minus 120572

0)

and 119865119862119866119869 119869119866 perp plane 119874119866119868

there4 119865119862 perp plane 119874119866119868

there4 119866119869 = 119874119866 times 119905119892120573

0=

ℎ

cos 1205740

times 119905119892120573

0

119865119862 = 119874119865 times 119905119892120573

0=

ℎ

cos (1205740minus 120572

0)

times 119905119892120573

0

∵ Δ119880119865119862 ∽ Δ119880119866119869

there4

119865119862

119866119869

=

119880119865

119880119866

there4

119865119862 minus 119866119869

119866119869

=

119880119865 minus 119880119866

119880119866

=

minus119866119865

119880119866

(A7)

there4 119880119866 =

119866119869 times 119866119865

119866119869 minus 119865119862

=

ℎ [119905119892120574

0minus 119905119892 (120574

0minus 120572

0)] cos (120574

0minus 120572

0)

cos (1205740minus 120572

0) minus cos 120574

0

(A8)

there4 119866119871 = 119874119866 times 119905119892120573 =

119874119868

cos 1205740

times 119905119892120573 =

ℎ

cos 1205740

times 119905119892120573 (A9)

InΔ119880119875119876 119866119871119875119876

=

119880119866

119880119876

there4 119875119876 =

119866119871 (119880119866 + 119866119876)

119880119866

(A10)


Thus

there4 119875

119909= 119875119876

=

ℎ

cos 1205740

times 119905119892120573

times(1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

(A11)




119909 119875119910) is

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times(1 +

(119905119892 (120574

0+ 120572) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A12)

Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10

12

3

4

5

6

7

89


2 Method



119864

119887= (120582 119879

119871) = 120576

119879

(120582 119879) sdot 119864

119887(119879

0 120582) (1)

119864


120576





LLower

LUpper

Measurement bar

Mold powder

Molten steel










pulled up


being pulled up

(a)


mpe

ratu

re (∘

C)0 50 100 150 200 250 300

200

400

600

800

1000

1200

1400

1600

02468101214161820222426


Tem

pera

ture

gra

dien

t (∘ C

mm

)

(b)






2 In

Measurement bar

Camera

Pulley

Steel chain


Pillar


Mechanical device

Measurement barTCF

Molten steel

Camera

Computer

Siemens S7-200



y

xP998400(px py)

H

W

ab

c d

g


119909

119901

119910



1to 1198752 the


1and 1198682


119909 119901


as (119875

119909 119875

119910 119875






(119875

119909 119875

119910 119875






120574


0


divides ang119870119874119869 ang119866119874119868 = 120574



119910) in


Y

X

A

EQ

P

K

G

L

J

D

F

C

IU

O

120572120573

1205740

B






120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times (1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

119875

119911= 0

(3)


0represents





1to ℎ2(ℎ

2gt ℎ

1) although ℎ

1and

ℎ


1minus ℎ





O2

O1

h1 h2

G1G2Q I

1205721

1205722


1

to ℎ2


1

to ℎ2


1

to 1198742


1

to 1198662









1and1198752have the


ℎ

1

cos 1205740

times 119905119892120573

1

times (1 +

(119905119892 (120574

0+ 120572

1) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

=

ℎ

2

cos 1205740

times 119905119892120573

2

times(1+

(119905119892 (120574

0+ 120572

2) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

ℎ

2minus ℎ

1= Δℎ

120572

1= arc119905119892

2119901

1199101times 119905119892120572

0

119867

120573

1= arc119905119892

2119901

1199091

times 119905119892120573

0

119882

120572

2= arc119905119892

2119901

1199102

times 119905119892120572

0

119867

120573

2= arc119905119892

2119901

1199092

times 119905119892120573

0

119882

(4)




119871 (119909 119910 120590) = 119866 (119909 119910 120590) lowast 119868 (119909 119910) (5)


119866 (119909 119910 120590) =

(12120587120590

2

) 119890

minus(119909

2+119910

2)

2120590

2

(6)


119863(119909 119910 120590) = (119866 (119909 119910 119896120590) minus 119866 (119909 119910 120590)) lowast 119868 (119909 119910)

= 119871 (119909 119910 119896120590) minus 119871 (119909 119910 120590)

(7)




119863 (119883) = 119863 +

120597119863

119879

120597119883

119883 +

1

2

119883

119879120597

2

119863

120597119883

2

119883(8)


119883 = minus

120597

2

119863

minus1

120597119883

2

120597119863

120597119883

119863 (

119883) = 119863 +

1

2

120597119863

119879

120597119883

119883

(9)



119867 = [

119863

119909119909119863

119909119910

119863

119909119910119863

119910119910

]

Tr (119867) = 119863119909119909+ 119863

119910119910= 120572 + 120573

Det (119867) = 119863119909119909119863

119910119910minus (119863

119909119910)

2

= 120572120573

(10)


119877 =

Tr (119867)2

Det (119867)=

(120572 + 120573)

2

120572120573

=

(119903120573 + 120573)

2

119903120573

2

=

(119903 + 1)

2

119903

(11)


2




2

+(119871 (119909 119910 + 1)minus119871 (119909 119910 minus 1))

2


119871 (119909 + 1 119910) minus 119871 (119909 minus 1 119910)

)

(12)






1to position

119875




119909 119871





119909 119871



119871






1




2







(a) (b)

(c)


in position 1198751

(b) image 119868

2

in position 1198752








1


119875






0 100 200 300 400 500 600 700 800 900 1000

1500

1400

1300

1200

1100

1000

900

800

700

Ly

d


sr

0 100 200 300 400 500 600 700 800 900 1000

30

28

26

24

22

20

18

16

14

Ly









119901

1199101

) and(119901

1199092 119901

1199102





Focus(mm)





25mm 964∘ 862∘ 819∘ 729∘

28mm 899∘ 798∘ 756∘ 668∘

36mm 757∘ 66∘ 622∘ 543∘

4mm 699∘ 607∘ 57∘ 495∘

6mm 50∘ 426∘ 398∘ 342∘

8mm 385∘ 326∘ 304∘ 26∘

12mm 262∘ 221∘ 205∘ 175∘

16mm 198∘ 166∘ 154∘ 132∘

30mm 106∘ 89∘ 83∘ 7∘

60mm 53∘ 45∘ 41∘ 35∘

100mm 32∘ 27∘ 25∘ 21∘

200mm 16∘ 13∘ 12∘ 11∘






(a) (b)


(a) (b)










Imagecoordinates

aftermovement


Actualdistance(mm)

Deviation(mm)





TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


4 Conclusion


Appendix






∵ Line 119869119866 perp line 119864119865


(A1)


119909 119875


















119910) As



119905119892120572 =

119866

1015840

119876

1015840

119874119866

1015840

=

119901

119910

119866

1015840

119864

1015840

119905119892 (ang119866

1015840

119874119864

1015840

)

=

119901

119910

(12)119867119905119892120572

0

=

2119901

119910times 119905119892120572

0

119867

(A2)

there4 120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

(A3)

Similarly

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A4)


119874119866 =

ℎ

cos 1205740

∵ 119868119866 = ℎ times 119905119892120574

0

119868119876 = ℎ times 119905119892 (120574

0+ 120572)

there4 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

(A5)

there4 119875

119910= 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0) (A6)













TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

464442

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


F998400

G998400

Q998400

E998400

O

h

IFGQEY

120572 1205740


119910



∵ 119868119866 = ℎ times 119905119892120574

0

119868119865 = ℎ times 119905119892 (120574

0minus 120572

0)


0minus 119905119892 (120574

0minus 120572

0))

∵ 119874119866 =

ℎ

cos 1205740

119874119865 =

ℎ

cos (1205740minus 120572

0)

and 119865119862119866119869 119869119866 perp plane 119874119866119868

there4 119865119862 perp plane 119874119866119868

there4 119866119869 = 119874119866 times 119905119892120573

0=

ℎ

cos 1205740

times 119905119892120573

0

119865119862 = 119874119865 times 119905119892120573

0=

ℎ

cos (1205740minus 120572

0)

times 119905119892120573

0

∵ Δ119880119865119862 ∽ Δ119880119866119869

there4

119865119862

119866119869

=

119880119865

119880119866

there4

119865119862 minus 119866119869

119866119869

=

119880119865 minus 119880119866

119880119866

=

minus119866119865

119880119866

(A7)

there4 119880119866 =

119866119869 times 119866119865

119866119869 minus 119865119862

=

ℎ [119905119892120574

0minus 119905119892 (120574

0minus 120572

0)] cos (120574

0minus 120572

0)

cos (1205740minus 120572

0) minus cos 120574

0

(A8)

there4 119866119871 = 119874119866 times 119905119892120573 =

119874119868

cos 1205740

times 119905119892120573 =

ℎ

cos 1205740

times 119905119892120573 (A9)

InΔ119880119875119876 119866119871119875119876

=

119880119866

119880119876

there4 119875119876 =

119866119871 (119880119866 + 119866119876)

119880119866

(A10)


Thus

there4 119875

119909= 119875119876

=

ℎ

cos 1205740

times 119905119892120573

times(1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

(A11)




119909 119875119910) is

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times(1 +

(119905119892 (120574

0+ 120572) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A12)

Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











pulled up


being pulled up

(a)


mpe

ratu

re (∘

C)0 50 100 150 200 250 300

200

400

600

800

1000

1200

1400

1600

02468101214161820222426


Tem

pera

ture

gra

dien

t (∘ C

mm

)

(b)






2 In

Measurement bar

Camera

Pulley

Steel chain


Pillar


Mechanical device

Measurement barTCF

Molten steel

Camera

Computer

Siemens S7-200



y

xP998400(px py)

H

W

ab

c d

g


119909

119901

119910



1to 1198752 the


1and 1198682


119909 119901


as (119875

119909 119875

119910 119875






(119875

119909 119875

119910 119875






120574


0


divides ang119870119874119869 ang119866119874119868 = 120574



119910) in


Y

X

A

EQ

P

K

G

L

J

D

F

C

IU

O

120572120573

1205740

B






120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times (1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

119875

119911= 0

(3)


0represents





1to ℎ2(ℎ

2gt ℎ

1) although ℎ

1and

ℎ


1minus ℎ





O2

O1

h1 h2

G1G2Q I

1205721

1205722


1

to ℎ2


1

to ℎ2


1

to 1198742


1

to 1198662









1and1198752have the


ℎ

1

cos 1205740

times 119905119892120573

1

times (1 +

(119905119892 (120574

0+ 120572

1) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

=

ℎ

2

cos 1205740

times 119905119892120573

2

times(1+

(119905119892 (120574

0+ 120572

2) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

ℎ

2minus ℎ

1= Δℎ

120572

1= arc119905119892

2119901

1199101times 119905119892120572

0

119867

120573

1= arc119905119892

2119901

1199091

times 119905119892120573

0

119882

120572

2= arc119905119892

2119901

1199102

times 119905119892120572

0

119867

120573

2= arc119905119892

2119901

1199092

times 119905119892120573

0

119882

(4)




119871 (119909 119910 120590) = 119866 (119909 119910 120590) lowast 119868 (119909 119910) (5)


119866 (119909 119910 120590) =

(12120587120590

2

) 119890

minus(119909

2+119910

2)

2120590

2

(6)


119863(119909 119910 120590) = (119866 (119909 119910 119896120590) minus 119866 (119909 119910 120590)) lowast 119868 (119909 119910)

= 119871 (119909 119910 119896120590) minus 119871 (119909 119910 120590)

(7)




119863 (119883) = 119863 +

120597119863

119879

120597119883

119883 +

1

2

119883

119879120597

2

119863

120597119883

2

119883(8)


119883 = minus

120597

2

119863

minus1

120597119883

2

120597119863

120597119883

119863 (

119883) = 119863 +

1

2

120597119863

119879

120597119883

119883

(9)



119867 = [

119863

119909119909119863

119909119910

119863

119909119910119863

119910119910

]

Tr (119867) = 119863119909119909+ 119863

119910119910= 120572 + 120573

Det (119867) = 119863119909119909119863

119910119910minus (119863

119909119910)

2

= 120572120573

(10)


119877 =

Tr (119867)2

Det (119867)=

(120572 + 120573)

2

120572120573

=

(119903120573 + 120573)

2

119903120573

2

=

(119903 + 1)

2

119903

(11)


2




2

+(119871 (119909 119910 + 1)minus119871 (119909 119910 minus 1))

2


119871 (119909 + 1 119910) minus 119871 (119909 minus 1 119910)

)

(12)






1to position

119875




119909 119871





119909 119871



119871






1




2







(a) (b)

(c)


in position 1198751

(b) image 119868

2

in position 1198752








1


119875






0 100 200 300 400 500 600 700 800 900 1000

1500

1400

1300

1200

1100

1000

900

800

700

Ly

d


sr

0 100 200 300 400 500 600 700 800 900 1000

30

28

26

24

22

20

18

16

14

Ly









119901

1199101

) and(119901

1199092 119901

1199102





Focus(mm)





25mm 964∘ 862∘ 819∘ 729∘

28mm 899∘ 798∘ 756∘ 668∘

36mm 757∘ 66∘ 622∘ 543∘

4mm 699∘ 607∘ 57∘ 495∘

6mm 50∘ 426∘ 398∘ 342∘

8mm 385∘ 326∘ 304∘ 26∘

12mm 262∘ 221∘ 205∘ 175∘

16mm 198∘ 166∘ 154∘ 132∘

30mm 106∘ 89∘ 83∘ 7∘

60mm 53∘ 45∘ 41∘ 35∘

100mm 32∘ 27∘ 25∘ 21∘

200mm 16∘ 13∘ 12∘ 11∘






(a) (b)


(a) (b)










Imagecoordinates

aftermovement


Actualdistance(mm)

Deviation(mm)





TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


4 Conclusion


Appendix






∵ Line 119869119866 perp line 119864119865


(A1)


119909 119875


















119910) As



119905119892120572 =

119866

1015840

119876

1015840

119874119866

1015840

=

119901

119910

119866

1015840

119864

1015840

119905119892 (ang119866

1015840

119874119864

1015840

)

=

119901

119910

(12)119867119905119892120572

0

=

2119901

119910times 119905119892120572

0

119867

(A2)

there4 120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

(A3)

Similarly

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A4)


119874119866 =

ℎ

cos 1205740

∵ 119868119866 = ℎ times 119905119892120574

0

119868119876 = ℎ times 119905119892 (120574

0+ 120572)

there4 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

(A5)

there4 119875

119910= 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0) (A6)













TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

464442

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


F998400

G998400

Q998400

E998400

O

h

IFGQEY

120572 1205740


119910



∵ 119868119866 = ℎ times 119905119892120574

0

119868119865 = ℎ times 119905119892 (120574

0minus 120572

0)


0minus 119905119892 (120574

0minus 120572

0))

∵ 119874119866 =

ℎ

cos 1205740

119874119865 =

ℎ

cos (1205740minus 120572

0)

and 119865119862119866119869 119869119866 perp plane 119874119866119868

there4 119865119862 perp plane 119874119866119868

there4 119866119869 = 119874119866 times 119905119892120573

0=

ℎ

cos 1205740

times 119905119892120573

0

119865119862 = 119874119865 times 119905119892120573

0=

ℎ

cos (1205740minus 120572

0)

times 119905119892120573

0

∵ Δ119880119865119862 ∽ Δ119880119866119869

there4

119865119862

119866119869

=

119880119865

119880119866

there4

119865119862 minus 119866119869

119866119869

=

119880119865 minus 119880119866

119880119866

=

minus119866119865

119880119866

(A7)

there4 119880119866 =

119866119869 times 119866119865

119866119869 minus 119865119862

=

ℎ [119905119892120574

0minus 119905119892 (120574

0minus 120572

0)] cos (120574

0minus 120572

0)

cos (1205740minus 120572

0) minus cos 120574

0

(A8)

there4 119866119871 = 119874119866 times 119905119892120573 =

119874119868

cos 1205740

times 119905119892120573 =

ℎ

cos 1205740

times 119905119892120573 (A9)

InΔ119880119875119876 119866119871119875119876

=

119880119866

119880119876

there4 119875119876 =

119866119871 (119880119866 + 119866119876)

119880119866

(A10)


Thus

there4 119875

119909= 119875119876

=

ℎ

cos 1205740

times 119905119892120573

times(1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

(A11)




119909 119875119910) is

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times(1 +

(119905119892 (120574

0+ 120572) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A12)

Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










y

xP998400(px py)

H

W

ab

c d

g


119909

119901

119910



1to 1198752 the


1and 1198682


119909 119901


as (119875

119909 119875

119910 119875






(119875

119909 119875

119910 119875






120574


0


divides ang119870119874119869 ang119866119874119868 = 120574



119910) in


Y

X

A

EQ

P

K

G

L

J

D

F

C

IU

O

120572120573

1205740

B






120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times (1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

119875

119911= 0

(3)


0represents





1to ℎ2(ℎ

2gt ℎ

1) although ℎ

1and

ℎ


1minus ℎ





O2

O1

h1 h2

G1G2Q I

1205721

1205722


1

to ℎ2


1

to ℎ2


1

to 1198742


1

to 1198662









1and1198752have the


ℎ

1

cos 1205740

times 119905119892120573

1

times (1 +

(119905119892 (120574

0+ 120572

1) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

=

ℎ

2

cos 1205740

times 119905119892120573

2

times(1+

(119905119892 (120574

0+ 120572

2) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

ℎ

2minus ℎ

1= Δℎ

120572

1= arc119905119892

2119901

1199101times 119905119892120572

0

119867

120573

1= arc119905119892

2119901

1199091

times 119905119892120573

0

119882

120572

2= arc119905119892

2119901

1199102

times 119905119892120572

0

119867

120573

2= arc119905119892

2119901

1199092

times 119905119892120573

0

119882

(4)




119871 (119909 119910 120590) = 119866 (119909 119910 120590) lowast 119868 (119909 119910) (5)


119866 (119909 119910 120590) =

(12120587120590

2

) 119890

minus(119909

2+119910

2)

2120590

2

(6)


119863(119909 119910 120590) = (119866 (119909 119910 119896120590) minus 119866 (119909 119910 120590)) lowast 119868 (119909 119910)

= 119871 (119909 119910 119896120590) minus 119871 (119909 119910 120590)

(7)




119863 (119883) = 119863 +

120597119863

119879

120597119883

119883 +

1

2

119883

119879120597

2

119863

120597119883

2

119883(8)


119883 = minus

120597

2

119863

minus1

120597119883

2

120597119863

120597119883

119863 (

119883) = 119863 +

1

2

120597119863

119879

120597119883

119883

(9)



119867 = [

119863

119909119909119863

119909119910

119863

119909119910119863

119910119910

]

Tr (119867) = 119863119909119909+ 119863

119910119910= 120572 + 120573

Det (119867) = 119863119909119909119863

119910119910minus (119863

119909119910)

2

= 120572120573

(10)


119877 =

Tr (119867)2

Det (119867)=

(120572 + 120573)

2

120572120573

=

(119903120573 + 120573)

2

119903120573

2

=

(119903 + 1)

2

119903

(11)


2




2

+(119871 (119909 119910 + 1)minus119871 (119909 119910 minus 1))

2


119871 (119909 + 1 119910) minus 119871 (119909 minus 1 119910)

)

(12)






1to position

119875




119909 119871





119909 119871



119871






1




2







(a) (b)

(c)


in position 1198751

(b) image 119868

2

in position 1198752








1


119875






0 100 200 300 400 500 600 700 800 900 1000

1500

1400

1300

1200

1100

1000

900

800

700

Ly

d


sr

0 100 200 300 400 500 600 700 800 900 1000

30

28

26

24

22

20

18

16

14

Ly









119901

1199101

) and(119901

1199092 119901

1199102





Focus(mm)





25mm 964∘ 862∘ 819∘ 729∘

28mm 899∘ 798∘ 756∘ 668∘

36mm 757∘ 66∘ 622∘ 543∘

4mm 699∘ 607∘ 57∘ 495∘

6mm 50∘ 426∘ 398∘ 342∘

8mm 385∘ 326∘ 304∘ 26∘

12mm 262∘ 221∘ 205∘ 175∘

16mm 198∘ 166∘ 154∘ 132∘

30mm 106∘ 89∘ 83∘ 7∘

60mm 53∘ 45∘ 41∘ 35∘

100mm 32∘ 27∘ 25∘ 21∘

200mm 16∘ 13∘ 12∘ 11∘






(a) (b)


(a) (b)










Imagecoordinates

aftermovement


Actualdistance(mm)

Deviation(mm)





TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


4 Conclusion


Appendix






∵ Line 119869119866 perp line 119864119865


(A1)


119909 119875


















119910) As



119905119892120572 =

119866

1015840

119876

1015840

119874119866

1015840

=

119901

119910

119866

1015840

119864

1015840

119905119892 (ang119866

1015840

119874119864

1015840

)

=

119901

119910

(12)119867119905119892120572

0

=

2119901

119910times 119905119892120572

0

119867

(A2)

there4 120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

(A3)

Similarly

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A4)


119874119866 =

ℎ

cos 1205740

∵ 119868119866 = ℎ times 119905119892120574

0

119868119876 = ℎ times 119905119892 (120574

0+ 120572)

there4 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

(A5)

there4 119875

119910= 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0) (A6)













TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

464442

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


F998400

G998400

Q998400

E998400

O

h

IFGQEY

120572 1205740


119910



∵ 119868119866 = ℎ times 119905119892120574

0

119868119865 = ℎ times 119905119892 (120574

0minus 120572

0)


0minus 119905119892 (120574

0minus 120572

0))

∵ 119874119866 =

ℎ

cos 1205740

119874119865 =

ℎ

cos (1205740minus 120572

0)

and 119865119862119866119869 119869119866 perp plane 119874119866119868

there4 119865119862 perp plane 119874119866119868

there4 119866119869 = 119874119866 times 119905119892120573

0=

ℎ

cos 1205740

times 119905119892120573

0

119865119862 = 119874119865 times 119905119892120573

0=

ℎ

cos (1205740minus 120572

0)

times 119905119892120573

0

∵ Δ119880119865119862 ∽ Δ119880119866119869

there4

119865119862

119866119869

=

119880119865

119880119866

there4

119865119862 minus 119866119869

119866119869

=

119880119865 minus 119880119866

119880119866

=

minus119866119865

119880119866

(A7)

there4 119880119866 =

119866119869 times 119866119865

119866119869 minus 119865119862

=

ℎ [119905119892120574

0minus 119905119892 (120574

0minus 120572

0)] cos (120574

0minus 120572

0)

cos (1205740minus 120572

0) minus cos 120574

0

(A8)

there4 119866119871 = 119874119866 times 119905119892120573 =

119874119868

cos 1205740

times 119905119892120573 =

ℎ

cos 1205740

times 119905119892120573 (A9)

InΔ119880119875119876 119866119871119875119876

=

119880119866

119880119876

there4 119875119876 =

119866119871 (119880119866 + 119866119876)

119880119866

(A10)


Thus

there4 119875

119909= 119875119876

=

ℎ

cos 1205740

times 119905119892120573

times(1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

(A11)




119909 119875119910) is

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times(1 +

(119905119892 (120574

0+ 120572) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A12)

Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










O2

O1

h1 h2

G1G2Q I

1205721

1205722


1

to ℎ2


1

to ℎ2


1

to 1198742


1

to 1198662









1and1198752have the


ℎ

1

cos 1205740

times 119905119892120573

1

times (1 +

(119905119892 (120574

0+ 120572

1) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

=

ℎ

2

cos 1205740

times 119905119892120573

2

times(1+

(119905119892 (120574

0+ 120572

2) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

ℎ

2minus ℎ

1= Δℎ

120572

1= arc119905119892

2119901

1199101times 119905119892120572

0

119867

120573

1= arc119905119892

2119901

1199091

times 119905119892120573

0

119882

120572

2= arc119905119892

2119901

1199102

times 119905119892120572

0

119867

120573

2= arc119905119892

2119901

1199092

times 119905119892120573

0

119882

(4)




119871 (119909 119910 120590) = 119866 (119909 119910 120590) lowast 119868 (119909 119910) (5)


119866 (119909 119910 120590) =

(12120587120590

2

) 119890

minus(119909

2+119910

2)

2120590

2

(6)


119863(119909 119910 120590) = (119866 (119909 119910 119896120590) minus 119866 (119909 119910 120590)) lowast 119868 (119909 119910)

= 119871 (119909 119910 119896120590) minus 119871 (119909 119910 120590)

(7)




119863 (119883) = 119863 +

120597119863

119879

120597119883

119883 +

1

2

119883

119879120597

2

119863

120597119883

2

119883(8)


119883 = minus

120597

2

119863

minus1

120597119883

2

120597119863

120597119883

119863 (

119883) = 119863 +

1

2

120597119863

119879

120597119883

119883

(9)



119867 = [

119863

119909119909119863

119909119910

119863

119909119910119863

119910119910

]

Tr (119867) = 119863119909119909+ 119863

119910119910= 120572 + 120573

Det (119867) = 119863119909119909119863

119910119910minus (119863

119909119910)

2

= 120572120573

(10)


119877 =

Tr (119867)2

Det (119867)=

(120572 + 120573)

2

120572120573

=

(119903120573 + 120573)

2

119903120573

2

=

(119903 + 1)

2

119903

(11)


2




2

+(119871 (119909 119910 + 1)minus119871 (119909 119910 minus 1))

2


119871 (119909 + 1 119910) minus 119871 (119909 minus 1 119910)

)

(12)






1to position

119875




119909 119871





119909 119871



119871






1




2







(a) (b)

(c)


in position 1198751

(b) image 119868

2

in position 1198752








1


119875






0 100 200 300 400 500 600 700 800 900 1000

1500

1400

1300

1200

1100

1000

900

800

700

Ly

d


sr

0 100 200 300 400 500 600 700 800 900 1000

30

28

26

24

22

20

18

16

14

Ly









119901

1199101

) and(119901

1199092 119901

1199102





Focus(mm)





25mm 964∘ 862∘ 819∘ 729∘

28mm 899∘ 798∘ 756∘ 668∘

36mm 757∘ 66∘ 622∘ 543∘

4mm 699∘ 607∘ 57∘ 495∘

6mm 50∘ 426∘ 398∘ 342∘

8mm 385∘ 326∘ 304∘ 26∘

12mm 262∘ 221∘ 205∘ 175∘

16mm 198∘ 166∘ 154∘ 132∘

30mm 106∘ 89∘ 83∘ 7∘

60mm 53∘ 45∘ 41∘ 35∘

100mm 32∘ 27∘ 25∘ 21∘

200mm 16∘ 13∘ 12∘ 11∘






(a) (b)


(a) (b)










Imagecoordinates

aftermovement


Actualdistance(mm)

Deviation(mm)





TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


4 Conclusion


Appendix






∵ Line 119869119866 perp line 119864119865


(A1)


119909 119875


















119910) As



119905119892120572 =

119866

1015840

119876

1015840

119874119866

1015840

=

119901

119910

119866

1015840

119864

1015840

119905119892 (ang119866

1015840

119874119864

1015840

)

=

119901

119910

(12)119867119905119892120572

0

=

2119901

119910times 119905119892120572

0

119867

(A2)

there4 120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

(A3)

Similarly

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A4)


119874119866 =

ℎ

cos 1205740

∵ 119868119866 = ℎ times 119905119892120574

0

119868119876 = ℎ times 119905119892 (120574

0+ 120572)

there4 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

(A5)

there4 119875

119910= 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0) (A6)













TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

464442

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


F998400

G998400

Q998400

E998400

O

h

IFGQEY

120572 1205740


119910



∵ 119868119866 = ℎ times 119905119892120574

0

119868119865 = ℎ times 119905119892 (120574

0minus 120572

0)


0minus 119905119892 (120574

0minus 120572

0))

∵ 119874119866 =

ℎ

cos 1205740

119874119865 =

ℎ

cos (1205740minus 120572

0)

and 119865119862119866119869 119869119866 perp plane 119874119866119868

there4 119865119862 perp plane 119874119866119868

there4 119866119869 = 119874119866 times 119905119892120573

0=

ℎ

cos 1205740

times 119905119892120573

0

119865119862 = 119874119865 times 119905119892120573

0=

ℎ

cos (1205740minus 120572

0)

times 119905119892120573

0

∵ Δ119880119865119862 ∽ Δ119880119866119869

there4

119865119862

119866119869

=

119880119865

119880119866

there4

119865119862 minus 119866119869

119866119869

=

119880119865 minus 119880119866

119880119866

=

minus119866119865

119880119866

(A7)

there4 119880119866 =

119866119869 times 119866119865

119866119869 minus 119865119862

=

ℎ [119905119892120574

0minus 119905119892 (120574

0minus 120572

0)] cos (120574

0minus 120572

0)

cos (1205740minus 120572

0) minus cos 120574

0

(A8)

there4 119866119871 = 119874119866 times 119905119892120573 =

119874119868

cos 1205740

times 119905119892120573 =

ℎ

cos 1205740

times 119905119892120573 (A9)

InΔ119880119875119876 119866119871119875119876

=

119880119866

119880119876

there4 119875119876 =

119866119871 (119880119866 + 119866119876)

119880119866

(A10)


Thus

there4 119875

119909= 119875119876

=

ℎ

cos 1205740

times 119905119892120573

times(1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

(A11)




119909 119875119910) is

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times(1 +

(119905119892 (120574

0+ 120572) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A12)

Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119867 = [

119863

119909119909119863

119909119910

119863

119909119910119863

119910119910

]

Tr (119867) = 119863119909119909+ 119863

119910119910= 120572 + 120573

Det (119867) = 119863119909119909119863

119910119910minus (119863

119909119910)

2

= 120572120573

(10)


119877 =

Tr (119867)2

Det (119867)=

(120572 + 120573)

2

120572120573

=

(119903120573 + 120573)

2

119903120573

2

=

(119903 + 1)

2

119903

(11)


2




2

+(119871 (119909 119910 + 1)minus119871 (119909 119910 minus 1))

2


119871 (119909 + 1 119910) minus 119871 (119909 minus 1 119910)

)

(12)






1to position

119875




119909 119871





119909 119871



119871






1




2







(a) (b)

(c)


in position 1198751

(b) image 119868

2

in position 1198752








1


119875






0 100 200 300 400 500 600 700 800 900 1000

1500

1400

1300

1200

1100

1000

900

800

700

Ly

d


sr

0 100 200 300 400 500 600 700 800 900 1000

30

28

26

24

22

20

18

16

14

Ly









119901

1199101

) and(119901

1199092 119901

1199102





Focus(mm)





25mm 964∘ 862∘ 819∘ 729∘

28mm 899∘ 798∘ 756∘ 668∘

36mm 757∘ 66∘ 622∘ 543∘

4mm 699∘ 607∘ 57∘ 495∘

6mm 50∘ 426∘ 398∘ 342∘

8mm 385∘ 326∘ 304∘ 26∘

12mm 262∘ 221∘ 205∘ 175∘

16mm 198∘ 166∘ 154∘ 132∘

30mm 106∘ 89∘ 83∘ 7∘

60mm 53∘ 45∘ 41∘ 35∘

100mm 32∘ 27∘ 25∘ 21∘

200mm 16∘ 13∘ 12∘ 11∘






(a) (b)


(a) (b)










Imagecoordinates

aftermovement


Actualdistance(mm)

Deviation(mm)





TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


4 Conclusion


Appendix






∵ Line 119869119866 perp line 119864119865


(A1)


119909 119875


















119910) As



119905119892120572 =

119866

1015840

119876

1015840

119874119866

1015840

=

119901

119910

119866

1015840

119864

1015840

119905119892 (ang119866

1015840

119874119864

1015840

)

=

119901

119910

(12)119867119905119892120572

0

=

2119901

119910times 119905119892120572

0

119867

(A2)

there4 120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

(A3)

Similarly

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A4)


119874119866 =

ℎ

cos 1205740

∵ 119868119866 = ℎ times 119905119892120574

0

119868119876 = ℎ times 119905119892 (120574

0+ 120572)

there4 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

(A5)

there4 119875

119910= 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0) (A6)













TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

464442

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


F998400

G998400

Q998400

E998400

O

h

IFGQEY

120572 1205740


119910



∵ 119868119866 = ℎ times 119905119892120574

0

119868119865 = ℎ times 119905119892 (120574

0minus 120572

0)


0minus 119905119892 (120574

0minus 120572

0))

∵ 119874119866 =

ℎ

cos 1205740

119874119865 =

ℎ

cos (1205740minus 120572

0)

and 119865119862119866119869 119869119866 perp plane 119874119866119868

there4 119865119862 perp plane 119874119866119868

there4 119866119869 = 119874119866 times 119905119892120573

0=

ℎ

cos 1205740

times 119905119892120573

0

119865119862 = 119874119865 times 119905119892120573

0=

ℎ

cos (1205740minus 120572

0)

times 119905119892120573

0

∵ Δ119880119865119862 ∽ Δ119880119866119869

there4

119865119862

119866119869

=

119880119865

119880119866

there4

119865119862 minus 119866119869

119866119869

=

119880119865 minus 119880119866

119880119866

=

minus119866119865

119880119866

(A7)

there4 119880119866 =

119866119869 times 119866119865

119866119869 minus 119865119862

=

ℎ [119905119892120574

0minus 119905119892 (120574

0minus 120572

0)] cos (120574

0minus 120572

0)

cos (1205740minus 120572

0) minus cos 120574

0

(A8)

there4 119866119871 = 119874119866 times 119905119892120573 =

119874119868

cos 1205740

times 119905119892120573 =

ℎ

cos 1205740

times 119905119892120573 (A9)

InΔ119880119875119876 119866119871119875119876

=

119880119866

119880119876

there4 119875119876 =

119866119871 (119880119866 + 119866119876)

119880119866

(A10)


Thus

there4 119875

119909= 119875119876

=

ℎ

cos 1205740

times 119905119892120573

times(1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

(A11)




119909 119875119910) is

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times(1 +

(119905119892 (120574

0+ 120572) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A12)

Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c)


in position 1198751

(b) image 119868

2

in position 1198752








1


119875






0 100 200 300 400 500 600 700 800 900 1000

1500

1400

1300

1200

1100

1000

900

800

700

Ly

d


sr

0 100 200 300 400 500 600 700 800 900 1000

30

28

26

24

22

20

18

16

14

Ly









119901

1199101

) and(119901

1199092 119901

1199102





Focus(mm)





25mm 964∘ 862∘ 819∘ 729∘

28mm 899∘ 798∘ 756∘ 668∘

36mm 757∘ 66∘ 622∘ 543∘

4mm 699∘ 607∘ 57∘ 495∘

6mm 50∘ 426∘ 398∘ 342∘

8mm 385∘ 326∘ 304∘ 26∘

12mm 262∘ 221∘ 205∘ 175∘

16mm 198∘ 166∘ 154∘ 132∘

30mm 106∘ 89∘ 83∘ 7∘

60mm 53∘ 45∘ 41∘ 35∘

100mm 32∘ 27∘ 25∘ 21∘

200mm 16∘ 13∘ 12∘ 11∘






(a) (b)


(a) (b)










Imagecoordinates

aftermovement


Actualdistance(mm)

Deviation(mm)





TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


4 Conclusion


Appendix






∵ Line 119869119866 perp line 119864119865


(A1)


119909 119875


















119910) As



119905119892120572 =

119866

1015840

119876

1015840

119874119866

1015840

=

119901

119910

119866

1015840

119864

1015840

119905119892 (ang119866

1015840

119874119864

1015840

)

=

119901

119910

(12)119867119905119892120572

0

=

2119901

119910times 119905119892120572

0

119867

(A2)

there4 120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

(A3)

Similarly

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A4)


119874119866 =

ℎ

cos 1205740

∵ 119868119866 = ℎ times 119905119892120574

0

119868119876 = ℎ times 119905119892 (120574

0+ 120572)

there4 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

(A5)

there4 119875

119910= 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0) (A6)













TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

464442

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


F998400

G998400

Q998400

E998400

O

h

IFGQEY

120572 1205740


119910



∵ 119868119866 = ℎ times 119905119892120574

0

119868119865 = ℎ times 119905119892 (120574

0minus 120572

0)


0minus 119905119892 (120574

0minus 120572

0))

∵ 119874119866 =

ℎ

cos 1205740

119874119865 =

ℎ

cos (1205740minus 120572

0)

and 119865119862119866119869 119869119866 perp plane 119874119866119868

there4 119865119862 perp plane 119874119866119868

there4 119866119869 = 119874119866 times 119905119892120573

0=

ℎ

cos 1205740

times 119905119892120573

0

119865119862 = 119874119865 times 119905119892120573

0=

ℎ

cos (1205740minus 120572

0)

times 119905119892120573

0

∵ Δ119880119865119862 ∽ Δ119880119866119869

there4

119865119862

119866119869

=

119880119865

119880119866

there4

119865119862 minus 119866119869

119866119869

=

119880119865 minus 119880119866

119880119866

=

minus119866119865

119880119866

(A7)

there4 119880119866 =

119866119869 times 119866119865

119866119869 minus 119865119862

=

ℎ [119905119892120574

0minus 119905119892 (120574

0minus 120572

0)] cos (120574

0minus 120572

0)

cos (1205740minus 120572

0) minus cos 120574

0

(A8)

there4 119866119871 = 119874119866 times 119905119892120573 =

119874119868

cos 1205740

times 119905119892120573 =

ℎ

cos 1205740

times 119905119892120573 (A9)

InΔ119880119875119876 119866119871119875119876

=

119880119866

119880119876

there4 119875119876 =

119866119871 (119880119866 + 119866119876)

119880119866

(A10)


Thus

there4 119875

119909= 119875119876

=

ℎ

cos 1205740

times 119905119892120573

times(1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

(A11)




119909 119875119910) is

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times(1 +

(119905119892 (120574

0+ 120572) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A12)

Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 100 200 300 400 500 600 700 800 900 1000

1500

1400

1300

1200

1100

1000

900

800

700

Ly

d


sr

0 100 200 300 400 500 600 700 800 900 1000

30

28

26

24

22

20

18

16

14

Ly









119901

1199101

) and(119901

1199092 119901

1199102





Focus(mm)





25mm 964∘ 862∘ 819∘ 729∘

28mm 899∘ 798∘ 756∘ 668∘

36mm 757∘ 66∘ 622∘ 543∘

4mm 699∘ 607∘ 57∘ 495∘

6mm 50∘ 426∘ 398∘ 342∘

8mm 385∘ 326∘ 304∘ 26∘

12mm 262∘ 221∘ 205∘ 175∘

16mm 198∘ 166∘ 154∘ 132∘

30mm 106∘ 89∘ 83∘ 7∘

60mm 53∘ 45∘ 41∘ 35∘

100mm 32∘ 27∘ 25∘ 21∘

200mm 16∘ 13∘ 12∘ 11∘






(a) (b)


(a) (b)










Imagecoordinates

aftermovement


Actualdistance(mm)

Deviation(mm)





TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


4 Conclusion


Appendix






∵ Line 119869119866 perp line 119864119865


(A1)


119909 119875


















119910) As



119905119892120572 =

119866

1015840

119876

1015840

119874119866

1015840

=

119901

119910

119866

1015840

119864

1015840

119905119892 (ang119866

1015840

119874119864

1015840

)

=

119901

119910

(12)119867119905119892120572

0

=

2119901

119910times 119905119892120572

0

119867

(A2)

there4 120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

(A3)

Similarly

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A4)


119874119866 =

ℎ

cos 1205740

∵ 119868119866 = ℎ times 119905119892120574

0

119868119876 = ℎ times 119905119892 (120574

0+ 120572)

there4 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

(A5)

there4 119875

119910= 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0) (A6)













TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

464442

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


F998400

G998400

Q998400

E998400

O

h

IFGQEY

120572 1205740


119910



∵ 119868119866 = ℎ times 119905119892120574

0

119868119865 = ℎ times 119905119892 (120574

0minus 120572

0)


0minus 119905119892 (120574

0minus 120572

0))

∵ 119874119866 =

ℎ

cos 1205740

119874119865 =

ℎ

cos (1205740minus 120572

0)

and 119865119862119866119869 119869119866 perp plane 119874119866119868

there4 119865119862 perp plane 119874119866119868

there4 119866119869 = 119874119866 times 119905119892120573

0=

ℎ

cos 1205740

times 119905119892120573

0

119865119862 = 119874119865 times 119905119892120573

0=

ℎ

cos (1205740minus 120572

0)

times 119905119892120573

0

∵ Δ119880119865119862 ∽ Δ119880119866119869

there4

119865119862

119866119869

=

119880119865

119880119866

there4

119865119862 minus 119866119869

119866119869

=

119880119865 minus 119880119866

119880119866

=

minus119866119865

119880119866

(A7)

there4 119880119866 =

119866119869 times 119866119865

119866119869 minus 119865119862

=

ℎ [119905119892120574

0minus 119905119892 (120574

0minus 120572

0)] cos (120574

0minus 120572

0)

cos (1205740minus 120572

0) minus cos 120574

0

(A8)

there4 119866119871 = 119874119866 times 119905119892120573 =

119874119868

cos 1205740

times 119905119892120573 =

ℎ

cos 1205740

times 119905119892120573 (A9)

InΔ119880119875119876 119866119871119875119876

=

119880119866

119880119876

there4 119875119876 =

119866119871 (119880119866 + 119866119876)

119880119866

(A10)


Thus

there4 119875

119909= 119875119876

=

ℎ

cos 1205740

times 119905119892120573

times(1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

(A11)




119909 119875119910) is

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times(1 +

(119905119892 (120574

0+ 120572) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A12)

Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)


(a) (b)










Imagecoordinates

aftermovement


Actualdistance(mm)

Deviation(mm)





TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


4 Conclusion


Appendix






∵ Line 119869119866 perp line 119864119865


(A1)


119909 119875


















119910) As



119905119892120572 =

119866

1015840

119876

1015840

119874119866

1015840

=

119901

119910

119866

1015840

119864

1015840

119905119892 (ang119866

1015840

119874119864

1015840

)

=

119901

119910

(12)119867119905119892120572

0

=

2119901

119910times 119905119892120572

0

119867

(A2)

there4 120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

(A3)

Similarly

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A4)


119874119866 =

ℎ

cos 1205740

∵ 119868119866 = ℎ times 119905119892120574

0

119868119876 = ℎ times 119905119892 (120574

0+ 120572)

there4 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

(A5)

there4 119875

119910= 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0) (A6)













TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

464442

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


F998400

G998400

Q998400

E998400

O

h

IFGQEY

120572 1205740


119910



∵ 119868119866 = ℎ times 119905119892120574

0

119868119865 = ℎ times 119905119892 (120574

0minus 120572

0)


0minus 119905119892 (120574

0minus 120572

0))

∵ 119874119866 =

ℎ

cos 1205740

119874119865 =

ℎ

cos (1205740minus 120572

0)

and 119865119862119866119869 119869119866 perp plane 119874119866119868

there4 119865119862 perp plane 119874119866119868

there4 119866119869 = 119874119866 times 119905119892120573

0=

ℎ

cos 1205740

times 119905119892120573

0

119865119862 = 119874119865 times 119905119892120573

0=

ℎ

cos (1205740minus 120572

0)

times 119905119892120573

0

∵ Δ119880119865119862 ∽ Δ119880119866119869

there4

119865119862

119866119869

=

119880119865

119880119866

there4

119865119862 minus 119866119869

119866119869

=

119880119865 minus 119880119866

119880119866

=

minus119866119865

119880119866

(A7)

there4 119880119866 =

119866119869 times 119866119865

119866119869 minus 119865119862

=

ℎ [119905119892120574

0minus 119905119892 (120574

0minus 120572

0)] cos (120574

0minus 120572

0)

cos (1205740minus 120572

0) minus cos 120574

0

(A8)

there4 119866119871 = 119874119866 times 119905119892120573 =

119874119868

cos 1205740

times 119905119892120573 =

ℎ

cos 1205740

times 119905119892120573 (A9)

InΔ119880119875119876 119866119871119875119876

=

119880119866

119880119876

there4 119875119876 =

119866119871 (119880119866 + 119866119876)

119880119866

(A10)


Thus

there4 119875

119909= 119875119876

=

ℎ

cos 1205740

times 119905119892120573

times(1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

(A11)




119909 119875119910) is

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times(1 +

(119905119892 (120574

0+ 120572) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A12)

Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Imagecoordinates

aftermovement


Actualdistance(mm)

Deviation(mm)





TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


4 Conclusion


Appendix






∵ Line 119869119866 perp line 119864119865


(A1)


119909 119875


















119910) As



119905119892120572 =

119866

1015840

119876

1015840

119874119866

1015840

=

119901

119910

119866

1015840

119864

1015840

119905119892 (ang119866

1015840

119874119864

1015840

)

=

119901

119910

(12)119867119905119892120572

0

=

2119901

119910times 119905119892120572

0

119867

(A2)

there4 120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

(A3)

Similarly

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A4)


119874119866 =

ℎ

cos 1205740

∵ 119868119866 = ℎ times 119905119892120574

0

119868119876 = ℎ times 119905119892 (120574

0+ 120572)

there4 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

(A5)

there4 119875

119910= 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0) (A6)













TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

464442

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


F998400

G998400

Q998400

E998400

O

h

IFGQEY

120572 1205740


119910



∵ 119868119866 = ℎ times 119905119892120574

0

119868119865 = ℎ times 119905119892 (120574

0minus 120572

0)


0minus 119905119892 (120574

0minus 120572

0))

∵ 119874119866 =

ℎ

cos 1205740

119874119865 =

ℎ

cos (1205740minus 120572

0)

and 119865119862119866119869 119869119866 perp plane 119874119866119868

there4 119865119862 perp plane 119874119866119868

there4 119866119869 = 119874119866 times 119905119892120573

0=

ℎ

cos 1205740

times 119905119892120573

0

119865119862 = 119874119865 times 119905119892120573

0=

ℎ

cos (1205740minus 120572

0)

times 119905119892120573

0

∵ Δ119880119865119862 ∽ Δ119880119866119869

there4

119865119862

119866119869

=

119880119865

119880119866

there4

119865119862 minus 119866119869

119866119869

=

119880119865 minus 119880119866

119880119866

=

minus119866119865

119880119866

(A7)

there4 119880119866 =

119866119869 times 119866119865

119866119869 minus 119865119862

=

ℎ [119905119892120574

0minus 119905119892 (120574

0minus 120572

0)] cos (120574

0minus 120572

0)

cos (1205740minus 120572

0) minus cos 120574

0

(A8)

there4 119866119871 = 119874119866 times 119905119892120573 =

119874119868

cos 1205740

times 119905119892120573 =

ℎ

cos 1205740

times 119905119892120573 (A9)

InΔ119880119875119876 119866119871119875119876

=

119880119866

119880119876

there4 119875119876 =

119866119871 (119880119866 + 119866119876)

119880119866

(A10)


Thus

there4 119875

119909= 119875119876

=

ℎ

cos 1205740

times 119905119892120573

times(1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

(A11)




119909 119875119910) is

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times(1 +

(119905119892 (120574

0+ 120572) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A12)

Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























119910) As



119905119892120572 =

119866

1015840

119876

1015840

119874119866

1015840

=

119901

119910

119866

1015840

119864

1015840

119905119892 (ang119866

1015840

119874119864

1015840

)

=

119901

119910

(12)119867119905119892120572

0

=

2119901

119910times 119905119892120572

0

119867

(A2)

there4 120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

(A3)

Similarly

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A4)


119874119866 =

ℎ

cos 1205740

∵ 119868119866 = ℎ times 119905119892120574

0

119868119876 = ℎ times 119905119892 (120574

0+ 120572)

there4 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

(A5)

there4 119875

119910= 119866119876 = 119868119876 minus 119868119866 = ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0) (A6)













TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

464442

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


F998400

G998400

Q998400

E998400

O

h

IFGQEY

120572 1205740


119910



∵ 119868119866 = ℎ times 119905119892120574

0

119868119865 = ℎ times 119905119892 (120574

0minus 120572

0)


0minus 119905119892 (120574

0minus 120572

0))

∵ 119874119866 =

ℎ

cos 1205740

119874119865 =

ℎ

cos (1205740minus 120572

0)

and 119865119862119866119869 119869119866 perp plane 119874119866119868

there4 119865119862 perp plane 119874119866119868

there4 119866119869 = 119874119866 times 119905119892120573

0=

ℎ

cos 1205740

times 119905119892120573

0

119865119862 = 119874119865 times 119905119892120573

0=

ℎ

cos (1205740minus 120572

0)

times 119905119892120573

0

∵ Δ119880119865119862 ∽ Δ119880119866119869

there4

119865119862

119866119869

=

119880119865

119880119866

there4

119865119862 minus 119866119869

119866119869

=

119880119865 minus 119880119866

119880119866

=

minus119866119865

119880119866

(A7)

there4 119880119866 =

119866119869 times 119866119865

119866119869 minus 119865119862

=

ℎ [119905119892120574

0minus 119905119892 (120574

0minus 120572

0)] cos (120574

0minus 120572

0)

cos (1205740minus 120572

0) minus cos 120574

0

(A8)

there4 119866119871 = 119874119866 times 119905119892120573 =

119874119868

cos 1205740

times 119905119892120573 =

ℎ

cos 1205740

times 119905119892120573 (A9)

InΔ119880119875119876 119866119871119875119876

=

119880119866

119880119876

there4 119875119876 =

119866119871 (119880119866 + 119866119876)

119880119866

(A10)


Thus

there4 119875

119909= 119875119876

=

ℎ

cos 1205740

times 119905119892120573

times(1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

(A11)




119909 119875119910) is

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times(1 +

(119905119892 (120574

0+ 120572) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A12)

Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















TCF

thic

knes

s (m

m)

0 1 2 3 4 5 6 7 8 9 10

464442

40

38

36

34

32

30

28

26

24

22

20

18

Time (hours)


F998400

G998400

Q998400

E998400

O

h

IFGQEY

120572 1205740


119910



∵ 119868119866 = ℎ times 119905119892120574

0

119868119865 = ℎ times 119905119892 (120574

0minus 120572

0)


0minus 119905119892 (120574

0minus 120572

0))

∵ 119874119866 =

ℎ

cos 1205740

119874119865 =

ℎ

cos (1205740minus 120572

0)

and 119865119862119866119869 119869119866 perp plane 119874119866119868

there4 119865119862 perp plane 119874119866119868

there4 119866119869 = 119874119866 times 119905119892120573

0=

ℎ

cos 1205740

times 119905119892120573

0

119865119862 = 119874119865 times 119905119892120573

0=

ℎ

cos (1205740minus 120572

0)

times 119905119892120573

0

∵ Δ119880119865119862 ∽ Δ119880119866119869

there4

119865119862

119866119869

=

119880119865

119880119866

there4

119865119862 minus 119866119869

119866119869

=

119880119865 minus 119880119866

119880119866

=

minus119866119865

119880119866

(A7)

there4 119880119866 =

119866119869 times 119866119865

119866119869 minus 119865119862

=

ℎ [119905119892120574

0minus 119905119892 (120574

0minus 120572

0)] cos (120574

0minus 120572

0)

cos (1205740minus 120572

0) minus cos 120574

0

(A8)

there4 119866119871 = 119874119866 times 119905119892120573 =

119874119868

cos 1205740

times 119905119892120573 =

ℎ

cos 1205740

times 119905119892120573 (A9)

InΔ119880119875119876 119866119871119875119876

=

119880119866

119880119876

there4 119875119876 =

119866119871 (119880119866 + 119866119876)

119880119866

(A10)


Thus

there4 119875

119909= 119875119876

=

ℎ

cos 1205740

times 119905119892120573

times(1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

(A11)




119909 119875119910) is

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times(1 +

(119905119892 (120574

0+ 120572) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A12)

Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










∵ 119868119866 = ℎ times 119905119892120574

0

119868119865 = ℎ times 119905119892 (120574

0minus 120572

0)


0minus 119905119892 (120574

0minus 120572

0))

∵ 119874119866 =

ℎ

cos 1205740

119874119865 =

ℎ

cos (1205740minus 120572

0)

and 119865119862119866119869 119869119866 perp plane 119874119866119868

there4 119865119862 perp plane 119874119866119868

there4 119866119869 = 119874119866 times 119905119892120573

0=

ℎ

cos 1205740

times 119905119892120573

0

119865119862 = 119874119865 times 119905119892120573

0=

ℎ

cos (1205740minus 120572

0)

times 119905119892120573

0

∵ Δ119880119865119862 ∽ Δ119880119866119869

there4

119865119862

119866119869

=

119880119865

119880119866

there4

119865119862 minus 119866119869

119866119869

=

119880119865 minus 119880119866

119880119866

=

minus119866119865

119880119866

(A7)

there4 119880119866 =

119866119869 times 119866119865

119866119869 minus 119865119862

=

ℎ [119905119892120574

0minus 119905119892 (120574

0minus 120572

0)] cos (120574

0minus 120572

0)

cos (1205740minus 120572

0) minus cos 120574

0

(A8)

there4 119866119871 = 119874119866 times 119905119892120573 =

119874119868

cos 1205740

times 119905119892120573 =

ℎ

cos 1205740

times 119905119892120573 (A9)

InΔ119880119875119876 119866119871119875119876

=

119880119866

119880119876

there4 119875119876 =

119866119871 (119880119866 + 119866119876)

119880119866

(A10)


Thus

there4 119875

119909= 119875119876

=

ℎ

cos 1205740

times 119905119892120573

times(1+

(119905119892 (120574

0+ 120572) minus 119905119892120574

0)times(cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0))timescos (120574

0minus 120572

0)

)

(A11)




119909 119875119910) is

119875

119909=

ℎ

cos 1205740

times 119905119892120573

times(1 +

(119905119892 (120574

0+ 120572) minus 119905119892120574

0) times (cos (120574

0minus 120572

0) minus cos 120574

0)

(119905119892120574

0minus 119905119892 (120574

0minus 120572

0)) times cos (120574

0minus 120572

0)

)

119875

119910= ℎ (119905119892 (120574

0+ 120572) minus 119905119892120574

0)

120572 = arc1199051198922119901

119910times 119905119892120572

0

119867

120573 = arc1199051198922119901

119909times 119905119892120573

0

119882

(A12)

Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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