43 M3 THE DERIVTION OF THE NAPPING EIUATIONS AND DIStMhTIOFORMULAE FOR THE S.. (U) DEFENSE NAPPING RGENCYHYOROORAPHIC/ TOPOORRPHIC CENTER LASNI.. 0 C RNOLD

UNCLSSIFIED 194 F0 223 N

mhoomohhmhilmhhmhhhhhhhhlmmhhhmhhhhhhhl

AI.

I

4

4

I

Woo

1IL25 LlAl Jll 1.61111-1 I- ...-

MICROCOPY RESOLUTION TEST CHART-

NATIONAL BUREAU OF STANDARDS-1963-A 13612.0

_p .

THE DERIVATION OF THE MAPPING EQUATIONS AND DISTORTIONp

FORMULAE FOR THE SATELLITE TRACKING MAP PROJECTIONS

00 byCaGregory Cote Arnold

Thesis submitted to the Faculty of the

IVirginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

in

GEODETIC ENGINEERING

APPROVED: 0 Og 8 4

><ADr. Steven D. Johnson

Dr. W ick J. Fell Prof. Eugene A. Taylor

June, 1984

CBlacksburg, Virginia

_LJ

/P4

UNCLASSIFIED

SE URtrY CLASSIFICATION OF TIIIS PA.E

REPORT DOCUMENTATION PAGE'Ia. REPORT SECURI r CLASSI FI CAI ION lb RESTRICTIVE MARKINGS

UNCLASSIFIED

2a. SECURITY CLASSIFICATION AUTHORITY 3. DISTRIBUTION/AVAI LABILITY OF REPORT

N/A Approved for public release;2b. OECLASSIF ICATIONiDOI INGRAii' G SCHEDULE distribution unlimited.

N/A__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

4. PERFORMING ORGANIZATION REPORT NUMBERIS) 5. MONITORING ORGANI7ATION REPORT NUMB3ER(S)

N/A N/A

6a. NAME OF PERFORMING ORGANIZATON fii OFFICE SYMBOt 7a, NAME OF MONi:ORING ORGAN!ZATIONDefense Mapping Agency I(fGpicIble)

Hydrographic/Topographic Ctr. DMAHTC/GST N/A6c. ADDRESS (,'.State an~d /11' Cude) 7b. ADDRESS (CitySil and il ~

6500 Brookes Lane &TE LE.TWashington, DC 20315 N/A ~UL099

Ba. NAME OF FUNUiNG'SPONSIIG 8b. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT !DEItrCA MEER

ORGA NI ZAT ION ( 'Ipirabk .t'--.

Defense Mapping AGency PAO ________________________Sc. ADDRESS ((Ctfy. Statt atid fi10~Il. SOURCE OF FUNDING NOS.Building 56, US Naval Observatory PROGRAM fPROJECT TASK* WK UNIT

Washington, DC 20315 ELMN NO1o oN

T& N/DeivtA I N/A N/A N/Aquato lle Disatortion Formulae for the

SaE~1llte'Tracking Map Projnef1ntir12 PERSONAL AUTHOR(S)

Arnol- Gerory S13a. 'YPE OF REPORT 13b, TIME COVERED 14. DATE OF REPORT (Yr. Mo . D-a.' 1 15, PAGE COUNT

FnlIF ROM TO 1984 6816. SUPPLE ME N T,4RY NOTAT ION This paper was submitted in partial fulfillment of the requirementsfor a Master of Science degree at V4-ginia Polytechnic Institute and University.

17 _ COSATI CUOES 18, SUBJECT TERMS (Continue on retera if LI'*rrS.Ur'. A-f 4lienttf' h- bN--k nw,i V,;

FIELJGOUPS~bOR.Conic Projection Satellite Tracking Projection08 02 __Cylindrical Projection Graticule

A BS T R ACT ( 0- 1 in, in .,,''rerj ,,sary and iden tik.' by bmlc number)

Rigorous derivations of the Satellite Tracking conic and cylindrical projections arepresented. The first fundamental quantities of the projection surface parameters asfunctions of spherical earth parameters are developed. Using these quantities, newly derivein this paper, general equations for the distortion in length, area, and azimuth aredeveloped. Examples of the graticule and distortion values are given for the Lan sat orbit.

21 ~I I I, VI 'LII IIA :.*4I ' ,ITLUNIA';I( u14t ~' U~~UNCLASSTFIrD13 022a__ _ __ _ ___F 84 7 1 kO.I (

ARNOLD, GREGORY C. (202) 227-21S2DA1TGSi

DD FORM 1473.83 APR 10 '1(;,N 01, 1 IArN 7I I,_ ,141.,) I C: I'' .' l VT I

7717775

THE DERIVATION OF THE MAPPING EQUATIONS AND DISTORTION

FORMULAE FOR THE SATELLITE TRACKING MAP PROJECTIONS

by

Gregory Cote Arnold

(ABSTRACT)

Rigorous derivations of the Satellite Tracking conic and cylindrical

projections are presented. The first fundamental quantities of the

projection surface parameters as functions of spherical earth para-

meters are developed. Using these quantities, newly derived in this

paper, general equations for the distortion in length, area, and

azimuth are developed. Examples of the graticule and distortion

values are given for the Landsat orbit.

TABLE OF CONTENTS

1. Introduction .................. ........1

1.2 Bjctive.nd.................................................2I

1.3 Sbjc e.................................................. 2

1.4 Satellite Apparent Longitude ............................... 3

2. Map Projection Concepts .................................o...... 5

2.1 The Projections ............................................ 5

2.2 Distortion on Map Projections .............................. 5

2.2.1 Length distortion ...................................... 6

2.2.2 Area distortion ........................................ 6

2.2.3 Angular distortion ..................................... 7

3. The Development of the Mapping Equations ...................... 8

3.1 Elements of the Satellite Orbit ............................ 8

3.1.1 Distance along the satellite track as a function of 0 .8

3.1.2 Distanice along the satellite track as a function of A..10

3.1.3 Satellite apparent longitude ........................... 10

3.1.4 The relationship between X' and * ................ 11

3.2 The Mapping Equations for the Cylindrical Projection ....... 13

3.2.1 The miapping equation for x ............................. 13

3.2.2 The mapping equation for y............................. 13

3.3 The Mapping Equations for the Conic Projection ............. 14

3.3.1 The mapping equation for 6.............................14

3.3.2 The mapping equation for P.......................o...... 16

3.3.3 The constant of the cone for a projection with two

conformal latitudes .................................... 18

3.3.4 The constant of the cone for a projection with one

conformal latitude.......... 0................ 19q

3.3.5 The constant of the cone when the conformal latitude

is the tracking limit ..................................20

3.3.6 The circle of groundtrack tangency ..................... 20

4. Distortion Analysis .......................................... 22

4.1 Cylindrical Projection Distortion Equations ............... 23

4.1.1 The first fundamental quantities ...................... 23

4.1.2 Scale distortion in the meridians and parallels ....... 24

4.1.3 Area distortion ....................................... 25

4.1.4 Angular distortion .................................... 25

4.2 Conic Projection Distortion Equations ..................... 25

4.2.1 The first fundamental quantities ...................... 25

4.2.2 Scale distortion In the meridians and parallels ....... 27

4.2.3 Area distortion ....................................... 27

4.2.4 Angular distortion .................................... 28

5. Results and Analysis ......................................... 29

5.1 The Cylindrical Projection ................................ 29-

5.2 The Conic Projection ...................................... 37

6. Conclusions .................................................. 44

References ....................................................... 45

Appendix A. A continuation of the derivation of the constant of - -

the cone for a projection with one conformal

latitude ............................................ 46

Appendix B. The derivation of the angular distortion equation... .47

Appendix C. Distortion tables ................................... 48

Addendix D. FORTRAN program and sample output ................... 53

Vita ............................................................. 63

IVI

LIST OF FIGURES

Figure 1. Elements of the satellite orbit ........................ 9

Figure 2. Elements of the conic projection to the plane ......... 15

Figure 3. Cylindrical projection meridian length distortion..30

Figure 4. Cylindrical projection parallel length distortion..32

Figure 5. Cylindrical projection area distortion ................ 33

Figure 6. Cylindrical projection azimuth distortionStandard parallel - 20 degrees ........................ 34

Figure 7. Cylindrical projection azimuth distortionStandard parallel - 45 degrees ........................ 35

Figure 8. Cylindrical projection azimuth distortionStandard parallel - 70 degrees ........................ 36

Figure 9. Conic projection meridian length distortion ........... 38

Figure 10. Conic projection parallel length distortion ........... 39

Figure 11. Conic projection area distortion ...................... 40

Figure 1.2. Conic projection azimuth distortionStandard parallel - 20 degrees ........................ 41

Figure 13. Conic projection azimuth distortionStandard parallel - 45 degrees ........................ 42

Figure 14. Conic projection azimuth distortion-Standard parallel - 70 degrees ........................ 43

Figure 15. Cylindrical projection graticule ...................... 61

Figure 16. Conic projection graticule ............................ 62

vS

LIST OF TABLES

Table 1. Cylindrical projection length and area distortionfor the Landsat orbit .................................... 49

Table 2. Conic projection length and area distortion for Ithe Landsat orbit ........................................ 50

Table 3. Cylindrical projection azimuth distortion for theLandsat orbit ............................................ 51

Table 4. Conic projection azimuth distortion for the Landsat 0orbit .................................................... 52

v

vi

1. INTRODUCTION

The growing use of remotely sensed earth imagery has prompted

interest in a new type of map projection, one in which some relation-

ship between satellite position and the earth is preserved during the

transformation, rather than the preservation of some feature or prop-

erty of the earth itself. A good example is the Space Oblique

Mercator described by Snyder (1978) on which conformality is preserved P

along a satellite groundtrack.

The Satellite Tracking projections have the property of portraying

all groundtracks of a particular satellite as straight lines. The

advantage of this is that, once the graticule is constructed, knowing

the location on the sphere of any subpoint allows the entire revolu-

tion to be drawn.

1.1 Background

In his article "Map Projections for Satellite Tracking", Snyder (1981)

introduces the cylindrical and conic Satellite Tracking projections.

The article briefly discusses the concept of satellite apparent

longitude from which the relationships between the revolving satellite

and the rotating earth are derived, and includes the development of

the mapping equations and examples of the graticules of both projec-

tions.

2

Although the mapping equations are developed, the complete relation-

ship between coordinates on the sphere and coordinates on the projection

surface is not. To define this relationship, and subsequently derive

general formulae for expressing various types of distortion on the

projection surface, the fundamental quantities of each projection must

be known. These are not determined by Snyder.

1.2 Objectivep

The objective of this thesis is the derivation of general distortion

formulae for the Satellite Tracking map projections. This is accom-

plished by first deriving the fundamental quantities of the projections

which relate the change in *,X on the sphere t'- the change in the pro-

jection parameters on the plane. From quantities, general equa-

tions for length, area, and angular distortion are derived. The equa-

tions used in the computation of the fundamental quantities and distor-

tion formulae are given in Pearson (1977).

1.3 Scope

After a brief explanation of terms and concepts as given in Richardus

and Adler (1972), a complete rederivation of the two projections is

presented. The derivations are included here for two reasons: first,

each step is completely explained, which should make them easier to

follow than the original; and second, the resulting mapping equationsL

are the basis for the determination of the distortion formulae.

In the chapter on distortion analysis, the fundamental quantities and

3

distortion equations are derived. Finally, numerical-examples using

the Landsat orbit are given.

Changes in the definition of some variables will affect the appearance

of several equations. Specifically, motion of the satellite along its

orbit will be calculated from the ascending node, the customary point B--

of reference, rather than the descending node. Also, Snyder refers to

a non-rotating earth which he later allows to rotate to complete his

description of satellite motion. Here, the non-rotating earth concept 0

is replaced by referencing the satellite to an astronomical coordinate

system.

0Three conditions are assumed in the derivation of the mapping equations

1. The earth is spherical and of uniform mass,

2. the orbit is circular, i.e., satellite velocity is constant,

3. the rotation axes of the selected astronomical and earth-fixed

coordinate systems are coincident.

Plotting the groundtrack is equivalent to orbit determination where

the above conditions serve as the math model of the orbit. Recogni-

zing the limitations of such a crude model, extrapolation on any

groundtrack should be restricted to the revolution containing the

known satellite subpoint.

1.4 Satellite Apparent Longitude

Two types of longitude are used in the derivation of the mapping

equations: first, earth-fixed, or geodetic, longitude (X), and second,

satellite apparent longitude (A'), defined in section 3.1.3. The

4

meridians on the projection represent geodetic longitude. However, in

order to compute the parallel spacing necessary to allow straight line

groundtracks, the motion of the satellite must be taken into consider-

ation. Therefore, the derivation of the mapping equations for * willbe based on X' instead of X.

I

2. MAP PROJECTION CONCEPTS

2. 1 The Projections

The transformations are applied to positions on a spherical earth of

radius R, defined in the latitude, longitude (O,X) coordinate system.

The two projection surfaces, the cylinder and the cone, are oriented

relative to the sphere in the normal aspect, i.e., their axes coincide

with the polar axis of the earth. The plane coordinate systems are

chosen such that the parametric lines of that system correspond with

the projection of the parametric lines of the sphere. On the cylinder,

where the graticule is composed of perpendicular straight lines, the

Cartesian (x,y) system is used. On the cone, where meridians are

straight lines converging at its apex, and parallels are arcs of concen-

tric circles, the (p,6) system is used. The line of contact between

the projection surface and the sphere is a parallel of latitude along

which scale is true. A secant cone or cylinder will have two standard

parallels, and a tangent surface will have one.

2.2 Distortion On Map Projections

Regardless of the type of transformation, all of the relationships

existing on the sphere cannot be duplicated on the plane. Any pro-

jection will contain distortions in distance, direction, size or

shape. The Satellite Tracking projections sacrifice some desirable

properties in order that satellite groundtracks are preserved as

straight lines.

5I

6

2.2.1 Length distortion

Length, or scale, distortion is a measure of the difference in ler

of a line on the projection surface compared with the length of tl

same line on a sphere which has been reduced to map scale. It is

usually expressed as a scale factor, the ratio of the two distance

Scale factor = distance on the plane

distance on the sphere

The scale factor along a standard parallel or meridian, where scal

true, is 1. A scale factor greater than I indicates an increase I

length on the projection; a value less than 1 indicates a decrease

in length.

On most projections, the scale factor is dependent on the azimuth

line. Formulae to be developed here are for distortion along the

meridians (Mm) and along the parallels (Mp).

Conformality is the property which maintains the shapes of differe

tially small areas on a projection. This means that, although the

magnitude of the scale distortion changes across the map, at any r

Mm = Mp.

This condition will be used in the derivation of the mapping equat

2.2.2 Area distortion

Area distortion (Da), also expressed as a ratio, compares an area

the map with the same area on the earth at map scale.

Da = area on the planearea on the sphere

7

Because the parametric curves of the two plane coordinate systems are

orthogonal, the area bounded by two differentially close pairs of

curves is the product of the lengths of the lines separating them.

Da = Mm Mp

2.2.3 Angular distortion

Angular distortion is measured by comparing an azimuth on the sphere to

its projected azimuth on the plane. For the Satellite Tracking pro-

jections, it is a function of azimuth and latitude. Angular distortion

will be zero, i.e., sphere azimuth equals plane azimuth, wherever the

condition of conformality is satisfied; otherwise, some distortion will

be present.

At each point on the sphere there exists a pair of perpendicular lines

that will remain perpendicular during the transformation; along these

lines, angular distortion is zero.

3. THE DEVELOPMENT OF THE MAPPING EQUATIONS

3.1 Elements Of The Satellite Orbit

Unlike projections which deal only with parameters of the earth-fixed --

coordinate system, the Satellite Tracking projections must take into

account the motions of satellite revolution and earth rotation. This

is because, under the above assumptions, the satellite orbital plane is

stationary with respect to the stars, with its earth orientation con-

stantly changing due to earth rotation. Because the transformation

between these two systems is a rotation about the nolar axis, only

longitude is affected.

To develop the projections, satellite motion along its orbit must be

defined in terms of earth-fixed coordinates (O,X). This is done by

first defining the orbit in astronomical coordinates ((D,A), and then

applying the transformation to (O,X). The elements of the orbit are

shown in Figure 1.

3.1.1 Distance along the satellite track as a function Of

Applying the sine law to triangle ACD in Figure 1 gives the relation-

ship between the angular distance traveled by the satellite (ax) as a

function of 0 and the inclination of the orbital plane (i).

sina = sinO (1)sini

Under the conditions imposed for the derivation, astronomical latitude

equals geodetic latitude. Therefore, ~ ,and equation (1) may be

8

9

|B

90-00

, A A D

AscendingF e E node

- Figure 1. Elements of the satellite orbit.

10

rewritten as

sina = sino (2)sint

3.1.2 Distance along the satellite track as a function of A

Applying the sine law to triangle ABC, and the cosine law to triangle

ACD in Figure 1 provide the following relationships. In both cases, A

is measured from the ascending node.

sina = cosO sinA (3)cos i

cosa = cost cosA (4)

Dividing equation (3) by (4) eliminates 0 and expresses a as a

function of A only.

tana = tanA (5)cost

3.1.3 Satellite apparent longitude

The geodetic longitude of a point beneath the satellite orbit is

bequal to the astronomical longitude plus the combined effects of sat-

ellite and earth motion on earth-fixed longitude (AA).

V - A + AX (6)

where the primed X indicates longitude of the satellite, known as

satellite apparent longitude. The angular distance along the orbit

covered over time t is

a - 2vt (7)

p

where p is the satellite period of revolution. During the same

interval, the earth-fixed longitude of the satellite subpoint changes by

AX = -2wt (8)P

where P is the earth period of rotation. AX is negative because the

earth rotates in the positive X direction, therefore, the longitude of

the point is always decreasing. Equate (7) and (8) to obtain

AX= - (9)P

Substitute equation (9) into (6) to obtain the final form of

satellite apparent longitude

X' = A - a (10)P

4

3.1.4 The relationship between A' and

Groundtracks on these projections will be constrained as straight lines -

by spacing the parallels as a linear function of the satellite apparent

longitude as computed in equation (10). To do this, dX'/do is required.

Differentiating equations (10), (2), and (5) 4

dX. = dA - da (11)dT d P do

da = cost (12)do sint cosa

dA = sec2 a cosi da (13)d sec1 A d$ *

* 4

12

Substituting (12) and (13) into (11)

d se = sec2a cosl - 2 coso

TO sec 2 A P /sini cosa

dX-(=cost ' Cosodo \cos a (1 + tan 2A) P) sini cosa

Reducing terms in the last equation:

cos 2a (I + tan 2A) = cos 2 a + cos 2a tan 2A

= cos 2 a + cos 2a tan 2 a cos 2 1 from (5)

= 1 - sin 2 sin 2i p

= 1 - sin sin 2 l from (2)sin't

= cos 2 op

-sin 2 \ from (2)sin cos = sin from(2)

= (sin2I - Cos2)

= (cos 2 - Cos21)

Continuing the derivation:

dX - (cos, - 2 cos

To- V P) (cos2o - cos7)0

W - cos - (B/P) cos2 (14)do- cosO (coso - cos )

Given dW and do on the sphere, the azimuth (A) of a great circle at

latitude * is computed as:

tanA - dA" cos (15)

TO

13

Substituting the expression for dX'/do from (14) gives the equation

expressing groundtrack azimuth on the sphere as a function of the orbit

parameters.

tanA - cosi - (p/P)cos 2 (16)(cos20 - cos 2 0),

3.2 The Mapping Equations For The Cylindrical Projection

Formulae for length distortion on the projection surface along the

meridians (Mm) and along the parallels (Mp) are defined as:

Mm - 1 dy (17a)R do

Mp 1 1 dx (17b)R coso d)

3.2.1 The mapping equation for x

To derive the mapping equation for x, define the scale to be true in

the x direction at 01, the conformal latitude.

Mp = 1

dx - R cos€ 1 dX

Integrating with the constraint that x-O when A-0 gives the mapping

equation for x.

x - RXcosol (18)

3.2.2 The mapping equation for y

To derive the parallel spacing, impose the condition of conformalitv

14

at

Mm -Mp

dy = dx (19)coso 1 dXN

Integrate equation (19) and substitute from (18) replacing X with V

to make y a linear function of the satellite apparent longitude.

dX'/do is a constant since it is evaluated at 01.

y - RX'cosol (20)

coso idx\

Because conformality is defined at 01, the groundtrack azimuth at 01

on the map (dx/dy) should equal the azimuth on the globe (A).

Rearranging equation (19), again using X' in place of X yields

dx -(dX coso 1dy \doll1

This agrees with the value of tanA in (15), therefore

(dV) tanA,do), coso,

Substituting into equation (20) gives the mapping equation for y.

y = RX'coso1 (21)tanA1

3.3 The Mapping Equations For The Conic Projection

3.3.1 The mapping equation for a

Figure 2 shows the elements of the projection to the plane in the (0,0)

N

15

Circle of ground-track tangency

IPo Co

Fc

Figure 2. Elements of the conic projection to the plane.

16

coordinate system. As with conventional conics in the normal aspect,

the mapping equation for 8 is

e = nX (22a)

In the derivation of p, the satellite apparent longitude is substituted

for earth-fixed longitude.

w = nA" (22b)where

n is the constant of the cone, and

8 is the angular separation of the meridians on the projection

surface.

3.3.2 The mapping equation for p

From the geometry of the plane triangle in Figure 2

angle C - 1800 - (e + S)

where S is the azimuth of the groundtrack at the equator on the pro-

jection surface. Applying the sine law gives a relationship between P

and p0, the radius from the apex of the cone to the equator.

p = posinS (23)sin(8 + S)

To obtain the mapping equation for p, evaluate the length po at a

conformal latitude and substitute into equation (23).

Formulae for length distortion in the (p,8) system are

Mm -I dp (24a)R dO

.. . . . . .. . . . . ... . . .. . ... . . . . . . .

17

Mp= p dO (24b)Rcoso dA

The minus sign in Mm indicates that p decreases as 0 increases.

Differentiating equation (22b) with respect to p and V yields

d6 = n (25)

dO = n dX" 0

Substituting the value of dX'/do from equation (15)

d= = n tanA (26)TO coso

Imposing conformality at 01 produces an expression for the length pl.

Mm = Mp

d= -cos /

Substituting p from equation (23) and dX'/dO from (25) yields D

p1 = -coso, i d_ ( p, sinSn do ksin(e + S)]O,

P1 = cosol pa sinS cos(6, + S) d (0 + S) 0n sin2 (61 + S) do

Substituting dO/do from equation (26)

p1 = po sinS tanA I

sin(e1 + S) tan(e 1 + S)

Comparing this with equation (23)

tanA1 - tan(6 1 + S)

18

A, =0e + s (27)

Substituting equation (27) into (23) evaluated at

p, = po sinS (28)sinA

Imposing true scale at # produces a second expression for p1.

Mp = 1

P1 = Rcosol dX

P1 = Rcos l (29)n

Substitute equation (29) into (28) to eliminate p1 and obtain the

equation for po.

po = RcosoI sinA, (30)n sinS

Substitute equation (30) into (23) to eliminate p0 and get the

mapping equation for p.

p = Rcoso, sinAI (31)n sin(e + S)

3.3.3 The constant of the cone for a projection with two confor-

mal latitudes

Equation (27) gives the value of e at the conformal latitudes and

02"

E1 A1 - S

19

62 A 2 - S

Substituting into equation (22b)

81 = nj = Al - S

62 = nA = A2 - S

Subtracting:

nX5 - nj = (A2 - S) - (A, - S) I

n = A2 - Al (32)

3.3.4 The constant of the cone for a projection with one conformal

latitude

As 02 approaches 01, equation (32) becomes

n = dA= dX \d ) 0 1

Differentiating equation (16) gives the expression for dA/do.D

sec 2A dA = {(cos 20 - cos 2 1) (2(p/P)sino coso)

+ (cosi - (p/P)(cos20)( (cos2 o - cos21)- (-2sino coso)}

(Cos 2 0 - cos 2 l)

dA = sin$ coso {(p/P)(cos 2o - 2cos 2i) + cosi}do (cos2 o - cos 2l) 3/2 (1 + tan2A)

Substituting this and the value of do/dXV from (15) into the equation

for n evaluated at 01 yields

n = sino1 coso, [(p/P) cos2 oi - 2cos 2i) + cosi} coso i

(cos40I - cos2t)I/ 2 (1 + tan 2A1 ) tanA1

20

n = sino, cos2 i {(p/P) cos 2 ] - 2cos 2 l) + cosl}(COS 2 0, - cos 2 i) 3 / 2 (tanA1 + tan

3A1 )

Expanding the denominator using the expression for tanA in (16)

(cos 24 1 - COS21) 3/2 (cot - (p/P) cos24l + (cos, - (p/P) cos 2 t,) 3

(COS2 0I - COSZI)' (COS2 I - COS21)3/2

Thus

n = sin ; cos 2 o, ((p/P) (cos2 01 - 2cos 2t) + cos1} ((cos 0l - cos41)(cost - (p/P)cosZ-1 )+(cost - (p/P)cos2 $) 3

The equation for n appearing in Snyder (1981) contains additional

expansions of the denominator. See Appendix A for a continuation

of the derivation.

3.3.5 The constant of the cone when the conformal latitude is thi

tracking limit

The tracking limit, the highest latitude reached by the satellite

equal to the inclination of the orbit. To determine the value of

replace 01 with 1 in equation (33).

n = sini cos 2t (cost - (p/P) cos 2 1)(cost - (p/P) cos 2t)O

n = sin((I - (p/P) cost) 2

3.3.6 The circle of groundtrack tangency

If the projection is to be constructed manually, the plotting of

groundtracks is made easier by computing the circle of tangency.

is a circle of radius ps to which all groundtracks are tangent.

21

From the geometry of the plane triangle in Figure 2

ps = po sinS

Substituting the value of p0o from equation (30)

Ps = Rcoso1 sinA l (35)n

The direction of the groundtracks is obtained from the sign of either

PS in equation (35) or S in equation (27). A negative sign indicates

a west azimuth when the satellite is travelling northward, or east

azimuth when it's going southward.

Equation (35) is needed to compute ps because equation (31) breaks

down when 0 equals the tracking limit. Because parallel spacing was

derived based on the satellite groundtrack, radii poleward of the

tracking limit are undefined, i.e., the sine of a computed in

equation (2) is greater than unity.

4. DISTORTION ANALYSIS 0

The Satellite Tracking projections were developed by defining the

relationship between parameters on the projection surface and para-

meters on the sphere at specific locations, i.e., conformality and

true scale at one or two latitudes. Now that the mapping equations

have been developed, general equations describing the distortion

properties of the projections can be computed from fundamental quan-

tities developed using the following transformation matrix.

E du2 2 du dv dv2 E"d02 do do d2

F du du du dv + du dv dv dv F"

do dX ddT dX d d de

G du2 2 du dv dv2 G,"T-Xd d2 " -T'2

where

u,v are the parametric quantities on the projection surface,

either (x,y) or (p,e),

O,X are the parameters on the sphere,

E', F', G' are the fundamental quantities of u,v with

respect to the projection surface,

E, F, G are the fundamental quantities of u,v with respect

to the sphere.

Also used in the distortion analysis are e, f, and g, the fundamental

quantities of o,X with respect to the sphere.

22

p

23

The following fundamental quantities for the sphere and plane (pro-

jection surface) are known through the equations defining differen-

tial length

cX on the sphere: ds2 = R2 (do)2 + R2cos2o (dX)2

e - R2 f = 0 g = R2cos 2o

x,y on the plane: dS2 = (dx)2 + (dy)2

E' = 1 F' = 0 G= 1

p,8 on the plane: dS2 = (dp)2 + p2(de)2

E' - 1 F* = 0 G p 2

4.1 Cylindrical Projection Distortion Equations

4.1.1 The first fundamental quantities

Recalling equations (15), (18), and (21), and evaluating their partials

dX' = tanA (15)

do coso

x = RXcoso1 (18)

y = R.'cosol (21)tanA1

dx = 0

dx = RcosoldX

dy= Rcos d. = Rcosol tanAdo tanA 1 do tanA1 coso

dy=dX

24

Substitute the known values into the transformation matrix.

E 0 0 R2cos2 1 tan 2A Itan2A1 cos2

F 0 R 2cos 2 1 tanA 0 0tanAl cos

G R2cos 2 1 0 0 1

E = R2COS20] tan 2A (36a)tan2A1 cos2o

F = 0 (36b)

G = R2 cos2 1 (36c)

These are the fundamental quantities of the cylindrical projection.

4.1.2 Scale distortion in the meridians and parallels

Mm (E\

R 2cos2 tan2A\

Mm cosol tanA (37)

tanA 1 cosO

= [Rcs,

Mp -cos4 (38)

COS(

I

25

4.1.3 Area distortion

Da - (EG_-_F2\eg- f2

/R 4COS4,b t an 2A

"( tan2A, COs24BR cos 0 i

D=co2 i tanA (9

Da Cos (39)tanAl cos 2

4.1.4 Angular distortion

A derivation of the w to 9 equation is given in Appendix B.

cosQ Vf V coso cotw(E coszo cotw +)

Rcosti tanA cost cotw EtanAi cost

IRZcos Zl tanzA cosz cotzw + RZcosZ¢O

tan2Al cos2 € /

cosSI tanA cotw (40)(tan2A cot2w + tanzAl)

4.2 Conic Projection Distortion Equations

4.2.1 The first fundamental quantities

Recalling equations (15), (22a), (22b), and (31) and evaluating

their partials:

d = tanA (15)de cos€

0 = nX (22a)

0 = nA' (22b)

26

p = Rcosol sinAl (31)n sin(e + S)

do=

dO = n

dX

42 = Rcosol sinAld I 1do n doksin(a + S)

- -Rcosol sinAi cos(e + S) d (8 + S)n sin2 (e + S) To

Because the mapping equation for p was developed using satellite

apparent longitude, the value for e in dp/do is gotten from equation

(22b). Substituting from (22b) and (15) yields

d (6 + S) = d (nX') + dS = n dX n tanAdo do dO do coso

42 = -Rcos~i sinAi tanAdo coso sin(e + S) tan(e + S)

dX

Substituting the partials into the transformation matrix

E R2 cos 2 tl sin2A1 tan 2A 0 0 1cos O sinz(e + S) tan2 (e + S)

-n Rcosol sinA 1 tanA

F 0 coso sin(e + S) tan(e + S) 0 0

G 0 0 n2 02LI J

R2 cos 2 t sin 2A1 tan2 A (41a)cos 2o sin (e + S) tan2 (e + S) -

27

F - 0 (41b)

C = n2P2 (41c)

Since p is known as a function of 0, substitute equation (31) into

(41c) to obtain the equation for G.

G = R2cos2$1 sin 2A1 (41d)sin2(e + S)

These are the fundamental quantities of the conic projection.

4.2.2 Scale distortion in the meridians and parallels

Mm = (E)

R2cos 2$1 sin 2A1 tan 2Akos€ sin(O + S) tan2(e + S)

\ R~cos RZ

Mm = cosl sinA tanA (42)coso sin(e + S) tan(e + S)

Mp =(G)

R.c4s2$1 in2AA

Mp = coso, sinAl (43)coso sin(e + S)

P4.2.3 Area distortion

Da = - F2

P

- = m,, . . . . m . . . . .i i . . . .. . .. . . . . . . . . .. . . . . . . . . . .

28

R4COS 4 $1 sin4Ai a2AI - cos7 sin4(o + S) tan2(8 + S).

Da = COS2t01 sinx2A, tanA (44)I cosl sin2(O + S) tan(e + S)

4.2.4 Angular distortion

cosQ2 A=s cI E cosic cotlw +G)

Rcosol s nA, tanA cos cotwcososin(2 + S tan( + S)

Rzcsz~ si'Altan~co'2Acotzw + RzcosT2 - sin A(coa1O siwnZ(6 + S) tan,2 (O + S) sinze+ ST)

cosil =tanA cotw 2 (45)(tan'A cot 'w + tan (a + S))

5. RESULTS AND ANALYSIS

Inspecting the distortion equations for both projections reveals

that they do satisfy the conditions of true scale and conformality

imposed during the derivation of the mapping equations.

Mml = Mp@I

Mm02 = MP0 2Mm =I

01

Mpol 1

01 '01I = j

02 W02

As a further check, Da = Mm Mp.

The following analysis is illustrated with plots of the various distor-

tions for the Landsat orbit. Tables containing the numerical distor-

tion values appear in Appendix C.

The parameters used in the calculations are

satellite revolution period = 103.267 minutes,

earth rotation period = 1440.0 minutes, P

satellite inclination = 99 092.

5.1 The Cylindrical Projection

The meridian scale distortion, Figure 3, is a function of the secant of

the latitude. However, because the spacing of the parallels is based on

29

IWO0

30

0 0 -

0 m

F- 0.

0 00-1-f

.Er -j

LO m N

J1iOH 7O

31

the satellite orbit, the effect of the orbit appears in the tanA term.

Since the azimuth of the groundtrack on the sphere increases with

latitude, the large range of Mm reflects the increased parallel spacing

required to introduce sufficient angular distortion to keep the projected

azimuth straight.

The 01 subscripted terms in all of the distortion equations allow the

scale factor to equal unity at the standard parallel.

The parallel scale distortion, Figure 4, is a function only of the secant

of the latitude. This indicates that latitude length is independent

of its projected distance from the equator because the meridians are

cast as equidistant vertical lines. Area distortion, the product of

Mm and Mp, is shown in Figure 5.

Figures 6, 7, and 8 show the effect of angular distortion at standard

parallels of 200, 450, and 700. Conformality insures that the azimuth

on the map equals the azimuth on the sphere at the standard parallel.

Because azimuth is a constant on the projection, but increases with

latitude on the sphere, constraining the groundtracks as straight lines

causes map azimuth to be greater than sphere azimuth below the standard

parallel and less than sphere azimuth above it. This effect can also

be seen in the scale distortions: Mm is less than Mp below 01 and

greater than Mp above 01 due to the relative sizes of A and Al in

equation (37).

ii l I I

32

oc

aI,

0 0

x F-

I- LI)

xLJ F

jjmim -j C'

o 0

Ui CL -

z(Jc II -l

~~37zOS

33

0

m00

00

U-)

z c 00

00

cc m

0a- 0

(-)

cm I-CM

z (i)

-0 4-

Lo NCu-

aOLDHJ 37HOS

34

CYLINDRICRL PROJECTION RZIMUTH DISTORTILRNDSAT ORBIT STRNDRRD PRRRLLEL = 2

90-4 01400 -

860 Go*--

700 Latitude

H60-1

50*/N I

I00-i

400 --

00 10 20 30 40 50 60 0 800 90SPHERE AZIMUTH

Figure 6. Cylindrical projection azimuth distortion.Standard parallel = 20 degrees.

35

CYLINDRICAL PROJECTION AZIMUTH DISTORTIONLRNDSRT ORBIT STRNDRRD PARRLLEL - 450 3

800 00

80--- -7 00 Latitde/

-, I60"

N 7 /40t./0/

a..

E30_ / / 320t./ 70°-

I I l I I I I I I I

0' 100 200 30' 40' 50' 60' 70' 80' 90'SPHERE AZIMUTH

Figure 7. Cylindrical projection azimuth distortion.

Standard parallel = 45 degrees.

. . . . .I

36

VA CYLINDRICAL PROJECTION AZIMUTH DISTORTIONLANDSAT ORBIT STANDARD PRALLEL 70

800*

7 00 Latitude

N

00

I W

0* 10200300o40*506000600900SPHERE AZIMUTH

Figure 8. Cylindrical projection azimuth distortion.Standard parallel = 70 degrees.

37

5.2 The Conic Projection

The equations for scale distortion are similar to those for the

cylindrical projection; biit here, the azimuth of the groundtrack

changes on the projection as well as on the sphere. This is reflected

in the addition of the sinA1/sin(e + S) term in both equations, which

relates the azimuth at the standard parallel, Al, to its general value

of (8 + S). Plots of the meridian and parallel scale distortions for

the tangent conic projection are shown in Figures 9 and 10; area dis-

tortion is shown in Figure 11.

0 Inspecting the tables of the scale distortion values in Appendix C

reveals that, for the tangent cone, Mm is greater than Mp everywhere

except at the standard parallel where they are equal. For this to be

true, comparing equations (42) and (43), the azimuth on the sphere, A,

must always be greater than or equal to the azimuth on the plane,

(6 + S). This is confirmed in the angular distortion plots in Figures

12, 13, and 14.

38

00

-. ........

U,

00"

00

0 IZ

0o

00

uZ C

01'zi

a. . cr u)

0 '4 cu Si

cu a

LO ~ -r ,r80-LOU 37U:S

39

_ _ _ _ __ _ _ __ a)

0Mo(I U,

00

z0

00

F--00

2M ILC ~ -

I-,7-1

0110 ca

LLI ILIm iio 0 .

i CC -M/ -j

/ 0:zr

U CL -J

NOIOUA. 37UOS

S 40

_ _ _ _ __ _ _ __ _ _ _ 0

0,

- -~ --C

LO 000 0

00

00

U F-.

0 M0 0

X IM2

u Cf) -jZ~aJZ

---- Inc

N~iOU 37H0

41

CONIC PROJECTION RZIMUTH DISTORTIONLRNDSAT ORBIT STANDRRD PARALLEL - 20*

9001 001 40 ° _

7001 Latitude

60

i 30 / /504

404

N00

00° • /!*0 4'*060 7'80 9'

r Figure 12. Conic projection azimuth distortion.Standard parallel = 20 degrees.

0 ° _

42

CONIC PROJECTION AZIMUTH DISTORTIONLRNDSRT ORBIT STRNDRRD PARLLEL = 450

9011 80400 -

B ot.- 60-*

40L ./

100

00 .7-

I 1 I I I I I 1 1 I

0 ° 10° 20 30 40 ° 50 ° 60 ° 70 80 ° 90 °

SPHERE AZIMUTH

Figure 13. Conic projection azimuth distortion.Standard parallel = 45 degrees.

Iv

43

CONIC PROJECTION AZIMUTH DISTORTION

LRNDSAT ORBIT STANDARD PARALLEL -70

904 _ 00

so G o*-.880 °

1 Booiu.700 Latitude

600-/

CL 400-1j CC<

20*-

I ot. ] I I

0

0 ° 10 °* 2 0 ° 3 0 ° 4 0 ° 5 0 ° 6 0 ° 70 ° 8 0 ° 9 0 °

SPHERE AZIMUTH

Figure 14. Conic projection azimuth distortion.Standard parallel = 70 degrees.

6. CONCLUSIONS

The purpose of this thesis is to derive general distortion formulae

for the Satellite Tracking map projections. The following conclu-

sions can be drawn:

1. Equations for length, area, and azimuth distortion for the

Satellite Tracking projections were derived.

2. Limitations in the use of these projections are now quanti-

tatively defined, e.g., the size, shape, and orientation of a

Landsat scene.

3. The cylindrical projection is preferred when the entire earth

is to be shown because it allows true scale to be defined both

above and below the equator.

4. The conic projection is preferred for depicting small areas

because conformality can be defined at any two parallels

within the area of interest.

Recommendations for further study:

These projections are valid only under the assumption of a spherical

earth and circular satellite orbit.

1. What errors in the plotted groundtrack are introduced as a

result of these assumptions.

2. Can the projections be modified to accept a non-circular orbit.

3. Can the transformation be made from an ellipsoidal earth.

44

p2

REFERENCES

Pearson, Frederick, "Map Projection Equations", Report TR-3624, NavalSurface Weapons Center, Dahlgren Laboratory, Dahlgren, VA, 1977.

Snyder, John P.,"The Space Oblique Mercator Projection", PhotogrammetricEngineering and Remote Sensing, Vol 44, No. 5, May 1978.

Snyder, John P.,"Map Projections for Satellite Tracking", PhotogrammetricEngineering and Remote Sensing, Vol 77, No. 2, February 1981.

Richardus, Peter and Adler, Ron K.,"Map Projections for Geodesists,Cartographers, and Geographers", North-Holland Publishing Company,Amsterdam, 1972.

45

Appendix A. A continuation of the derivation of the constant of the

cone for a projection with one conformal latitude.

Recalling equation (33):

n =sini cos 2cbi {(p/p)(COS2 in - 2cos 2 l) + COS,}

(co2 ~ -COS2 1)(Cost - (p/p)COS2 1) + (COS, - (p/p)COS2 1)

3

Expanding the denominator:

(cos2 1 cos2 1)(cos, _ (p/p)cos241) +

(cos3t 3(p/p)cos2 p1 cos2 1 + 3(p/p)2 cos' b1 COS, (p) 3 cos6p1)

= -2(p/P)cos 2 p1 cos2 1 + 3(p/p)2COS4 1 cost

(p/P)3 cos6 1 + cos 2 1 COS, - (p/p)COS4?1

= cos2 W1 -2(p/P)cos2 I + 3(p/p)2 cos 2 1 cost

(p/P)3 cos4 1 + COSt - (p/p)cos2 1 )

= cos2c 1 (-(p/p)COS2%1 f(p/p)2 cos2 1 - 2(p/P)cost + 1.1 +

COS, f(p/p)2 cos241 - 2(p/P)cost + 1)

= cos2 1 ((COSt - (p/p)cos2c ,){(p/p) 2 cos 2 , - 2 (p/P)cosI + 1})

= COS2c 1 ((COSt - (p/p)cos2 1)f(p/p) ((p/p)cos2 1 - 2cosl) + 1})

Substituting this back into equation (33) yields the final form of n.

n = sin~1 f(p/P) (cos2 $1 - 2cos2t) + cos,)(cos - p/Pcos~ 1)(p/p)((p/p)COS

2 1 - 2cosl) + 11

This form is useful to test the condition where =900 and the

projection becomes azimuthal. This could not be done using equation

(33) because of the cos2 , term in the numerator.

46

Appendix B. The Derivation of the Angular Distortion Equation

/S

GdX R Lcosc di

Projection surface Spherical earth

On tte projection surface:

cosQ dodS

VE (dO)" + G (di)

I

coso V /- d_

On the sphere:

ds cosw = R d

ds sinw = R cos4 dX

do = R cos4 coswdX R sinw

d = cost cotwdX

Substituting:

cos J = cos cot'E cos2'- cot2w + G

47

II

P

Appendix C. Distortion Tables.

Table 1. Cylindrical projection length and area distortion for thLandsat orbit.

Table 2. Conic projection length and area distortion for theLandsat orbit.

Table 3. Cylindrical projection azimuth distortion for the Landsaorbit.

Table 4. Conic projection azimuth distortion for the Landsat orbi

8

0

48

49

Table 1. Cylindrical projection length and area distortion for theLandsat orbit.

Standard 0Parallel QO400 600 800

0 Meridian Length Distortion0 1.0 .66762 .31363 .01816

10 1.02179 .68217 .32047 .0185520 1.09298 .72969 .34279 .0198530 1.23456 .82421 .38720 .0224240 1.49787 1.0 .46978 .0272050 2.01389 1.34451 .63162 .0365760 3.18846 2.12866 1.0 .0579070 6.89443 4.60283 2.16231 .1251980 55.07144 36.76658 17.27213 1.0 480.908ccO

Parallel Length Distortion

0 1.0 .76604 .5 .1736510 1.01543 .77786 .50771 .1763320 1.06418 .81521 .53209 .1847930 1.15470 .88455 .57735 .2005140 1.30541 1.0 .65270 .2266850 1.55572 1.19175 .77786 .2701560 2.0 1.53209 1.0 .3473070 2.92380 2.23976 1.46190 .5077180 5.75877 4.41147 2.87939 1.080.908 6.32830 4.84776 3.16415 1.09890

Area Distortion

0 1.0 .51142 .15682 .0031510 1.03756 .53063 .16270 .0032720 1.16a12 .59485 .18240 .0036730 1.42555 .72906 .22355 .00449 440 1.95533 1.0 .30663 .0061750 3.13306 1.60232 *.49131 .0098860 6.37691 3.26130 1.*0 .0201170 20.15798 10.30926 3.16109 .0635680 317.14380 162.19480 49.73312 1.080.908onm0w

IL *

50

Table 2. Conic projection length and area distortion for theLandsat orbit.

StandardParallel 100 300 500 700

* Meridian Length Distortion

001.00313 1.07991 1.71646 23.26322

10 1.0 1.00398 1.35754 6.9207920 1.04275 .97729 1.14552 3.3712630 1.14572 1.0 1.02572 2.0649040 1.34649 1.08700 .97692 1.4612950 1.73949 1.28047 1.0 1.1571060 2.60194 1.69556 1.12369 1.0167970 5.08699 2.73190 1.44871 1.080 29.00155 9.41190 3.09745 1.2286480.908 c 0c

Parallel Length Distortion

0 .99671 .99621 1.17680 4.0574010 1.0 .96599 1.05324 2.2253220 1.03510 .96693 .98251 1.5764730 1.10816 1.0 .95174 1.2629440 1.23355 1.07194 .95670 1.0948950 1.44188 1.19859 1.0 1.0082460 1.80333 1.41446 1.09269 .9786170 2.51103 1.79895 1.25919 1.080 4.20804 2.42253 1.47016 1.0667680.908 4.23325 2.24930 1.35775 1.04704

Area Distortion

0 .99983 1.07582 2.01993 94.3881010 1.0 .96984 1.42982 15.4009520 1.07935 .94497 1.12548 5.3146930 1.26964 1.0 .97622 2.6078440 1.66097 1.16520 .93462 1.5999550 2.50814 1.53477 1.0 1.1666360 4.69216 2.39830 1.22784 .9950470 12.77360 4.91455 1.82420 1.080 122.03977 22.80060 4.55374 1.3106680.908 Oc

51

Table 3. Cylindrical projection azimuth distortion for the Landsatorbit.

Latitude 00 400 600 800

Sphere Map Azimuth

Azimuth Standard Parallel =200

0.0 0 o0 000 000 0 0

10 10.26495 8.96897 6.48077 1.0843620 20.49676 18.04498 13.19640 2.23744

30 30.66688 27.32916 20.40258 3.5464340 40. 75502 36.90925 28.39457 5. 1469550 50.75153 46.84937 37.51585 7.29022

60 60.65803 57.17734 48.13385 10.53262

70 70.48670 67.87198 60.53498 16.4323580 80.25838 78.85585 74.69298 31.33252

90 90.0 90.0 90.0 90.0

Standard Parallel =450

0 0.0 0.0 0.0 0.010 12.03481 10.52545 7.61675 1.27646

20 23.75263 20.98287 15.43133 2.6334330 34.91723 31.31478 23.64675 4. 17284

40 45.41324 41.48222 32.47151 6.0527850 55.23919 51.46883 42. 10802 8.56439

60 64.47477 61.28085 52.71885 12.3460170 73.24646 70.94427 64.36273 19. 14672

80 81.70266 80.50024 76.91131 35.62808

90 90.0 90.0 90.0 90.0

Standard Parallel =700

0 0.0 0.0 0.0 0.010 22.57733 19.91899 14.61773 2.48838

20 40.63871 36.79633 28.29625 5.1259730 53.70224 49.87562 40.49658 8.09850

40 63.18850 59.89055 51.14122 11.68437

50 70.41251 67.78976 60.43412 16.3686260 76.24248 74.30808 68.67779 23.11683

70 81.22568 79.95673 76.17587 34.10376

80 85.72366 85.09605 83.20187 54.41874

90 90.0 90.0 90.0 90.0

52

Table 4. Conic projection azimuth distortion for the Landsat orbit.

Latitude 0O 400 600 800

Sphere Map Azimuth

Azimuth Standard Parallel =200

00 0? 0?0 0?0o?10 9.71601 9.55806 7.64103 1.9524820 19.46493 19.16619 15.47873 4.0251930 29.27632 28.86987 23.71462 6.3691440 39.17314 38.70529 32.55515 9.2147250 49.16891 48.69490 42.19983 12.9750260 59.26561 58.84354 52.80775 18.5140570 69.45275 69.13623 64.43463 27.9766580 79.70805 79.53849 76.95195 47.6344890 90.0 90.0 90.0 90.0

Standard Parallel =450

0 0.0 0.0 0.0 0.010 7.70744 9.95383 9.44635 4.0861120 15.60828 19.91319 18.95443 8.3883330 23.89994 29.88297 28.58072 13.1652340 32.78316 39.86682 38.37095 18.7756250 42.44963 49.86671 48.35445 25.7725260 53.04910 59.88269 58.53894 35.0582470 64.62944 69.91288 68.90689 48.0640880 77.06188 79.95362 79.41530 66.4801490 90.0 90.0 90.0 90.0

Standard Parallel 700

0 0.0 0.0 0.0 0.0

10 1.76150 7.52606 9.63174 8.7041520 3.63232 15.25417 19.30566 17.5373930 5.75013 23.39270 29.05984 26.6238440 8.32610 32.15784 38.92427 36.0750850 11.74211 41.76285 48.91710 45.9780860 16.80910 52.38369 59.04168 56.3776870 25.60347 64.09087 69.28501 67.2565380 44.68725 76.75732 79.61826 78.52028

90 90.0 90.0 90.0 90.0

Appendix D. FORTRAN program and sample output.

This program computes values for constructing the graticules of the

Satellite Tracking projections. The type of the projection is deter- _

mined by the values in the assignment statements in the "initialize

variables" section. For example, if PHI1 equals zero, or PHI1 equals

-PH12, a cylindrical projection is produced; otherwise the result is a

conic.

Output is included for cylindrical and tangent conic projections with a

standard parallel of 300. Plots of the graticules are shown in Figures

15 and 16.

5

53

54

z ~ ~ a -Ca. .acC

4u 1- *0

X _j O.-C ZZO 0 0-b--

(M uO a i - IL $m 0 to X w w AuILwUW

4a.J4 " -. & w 6- 4 4M W ( 40 a - 04 1- mP.-p.t- WI- Zm (A m _j o az'-. um

UX M o 0 .- U4YI1- 0 a z ZUWO01. -- W Ui .9b- JC=i -b t-~ a.mIa wZ. CEw m K301

m -Mq -Z Ix w~ m. 0 0j )0I- M

tfo'j 4 1Ja fm<.& I I- W~U *JW.LWJ I0" t-= _j - WA41- 4 Xb. w -C

0 0-0 P. 3 i- -C 0 ~&'-4'-0 M:

w ~ ~ ul.9 U 00 ej U 0 ot- -0 OO . a. W-) M www.im4x -W < .JZOX'-IUL

P__-CIx0 ==U O.I-1--X $.- 0 00-iwmzU) 7 I Za- I-fnlo -40' P0 *2 Z Z-C0400-

0.jo0a. 0 - 1-- CL P-.0 =4 ~ .~.~I < z

Ucz'- I- (S0U0 401160-W mmzIML&LZ

M -W &a444 z 0 . Fm-Z X..L 4 1< -Cu (Aa~I %D0I-- (X-LLiiWxx a.. 4

C2-c.Z z-= &aJZ "4~~ - 10 U_ Lh.. -. O

CL. (A w =C040-bs- op- lao Zoomp~-t-100UUXX-4Z~ La = 0 - Li:)

_i O-zU sIiiow wz i-i=

wo wx U0t-- I.- U < W Ii * I f II I I IaI I II II

-i I- M 0 4 4I 0~- ~LA cL - m Pl-U I-- "AO CL~t m --

0 CL 0 m >0m& mm 0 O.a CaU I.-Z 0 w a m -M-IE j..

55

(LI

An a A

0 4 a u I a

00 04 to

I. r 0 4%,*-oi i m0 U W 0

* -0 04

~~C 00 = 0/ 0

(n 0 4 w - U0. &zSW C (n - aj -

0 ZZ 0W~ Z 0

Zf~h 0 1. 1.- - 1.-

-0.Z .4 w" 0 uZZWIsots" NNILa = to 0)a0 CO m

h.~ 0 41 S Xf fOW *~~ S - Z ZZ Z l..X.

C.00 *0 -J b---~ = Q 4Z 44 w C (L I-- -s. uZ0r

*UM =~I L" x * c I - 'ft = = w 4.4c "i %C ~ 'a O4 (A 0O4 %0 .

(it iJ =r a m g

56

a.a-

I- x

CL W)-* 0

-b P- 4 1 .CL

0.. Z UN 0I)C z~ (N .0 W

I. .tj (L4 ACI-- P0 C- *. £1.

M. us 4 -c -

a4 z cu W a W M

Loo* 0o (mS x M = -

AA I. I ( z 0 w0.c-- 0. Z - th-s'* *

0 oc CzC = 0 xa (ON 4 4c< VN 49 0.. - -

ecOK O C a 000i-Oll &S* 0 0 . 440 uL . -a~

vu I. w-- -3-* a

-m . U. N u * zan~~~~I a-O ~ ~ - X a3W

uN IN in 9 . C3LLU, 1*- _0 * -j ~-4' W W -C _ - I.- I-- 0 ZL 2C

~0 Z x 4

57

a -

m faa

z 4 0

L. z 0e 0

4 4c a lo 49-~P- "- 0 00

cc 0. =

= = 0Lu0 L Z 0- W 0a 0A

t" ~ ~ I i is J a a * iL L

to a~ = Co a0)0 Z* 0 0 "- -Z

m1 1a -Amw0 0..m =0. F"$- A 1 3 0 -- 1- % i-u >

4 L 44 000-0* =0 z4 0

o N0 x 0 .J C x 0 a m tP-3- w. x x w M- I.- -l 0m m

CL CL j Cx - zu- I-n jt

4o oszo u *

585

C58

rI

- L -: X £ .a . .- q

0 m-A -W

-. U. x

u. 49 - jUj

4c C .q9 o C S40 9-- m. I-a-4

4i Ni- D r - M Cz. -Z -W IA caJ 0- .

ccI. Z C ZjeI- Lo 0 0--MZz t

Q -4

P. -- A x - - - - - -

. -& 0.* z0 U.I- . I-I. .- f0 i 0 c-9 -C 49A 4

z I. a 0000Za. 00 zoj V7 U. .'0 W . . w 60 42 CD -. ~)

N LU 4.0

59

CYLINOICAL PROJECTION

STANnAPO PARALLEL =30.00

LATITUOE y000 6&3000

10.00 .1423920,00 .291213n1O00 .45470*.I 00o .06459150.00 .6897960.00 i,2448970.00 1.89!80000 4.3341780.91 5.86098

LONGITUDE X.00 .00000

10.00 .1511520.00 .302303n.oo .4.34540.00 .0046050.00 .75575bn.l0 .906907n.oo 1.'5a0580.00 1.292090,00 1.36035

Ii

60

II

CONSTANT OF THE CONE ..24794

pAnius OF CIRCLE OF GROUNOTRACK TANGENCY = -. 84314

EQUATOR GROUNOTRACK AZIMUTH ON THE MAP - 12.11332

STAJDARO PARALLEL 1 2 30.00

STA1G4AOR PARALLEL 2 a 3u.O0

LATITUDE RHO.00 4.01791

10.00 3.8368320.00 3.6646130900 3.4928400.00 3.3!185

50600 3.107336n.O0 2.d523970.00 2.49152a0.00 1.6966360.91 1.43346

LONITUOE THETA.00 orqooo

1000 2.4794320900 4,9;886300no 7.43829%0.00 9.9177250.00 12.3q71560000 14sa765770,00 17.35600,.0a 19.3543

9no0 22.31486

. . .. .. . . ... ll ll 1" - i . .. . .. . . .. . .. . . . .

61

I 180* -901 00 900 18e00

800

600

CYLINDRICAIL PROJECTION

STANDARD PRALLEL = 300

Figure 13. Cylindrical projection graticule.

00

0 LA0

-

w

0~

z0

F-

LI

00CD~

z0U

CD

VITA 4

Gregory Arnold was born in Reading, Pa, August 8, 1950. He received

a B.A. in Geology from Lehigh University in 1972 and a B.S. in

Cartography from George Washington University in 1982.

He served in the U.S. Army from 1972 to 1975. Since then he's worked

as a Cartographer for NOAA and the Geological Survey. He is now

a Physical Scientist working for the Defense Mapping Agency.

63

IbI

4As.

Ir

AA

* 44

A' * *

*~tA

-f 7v

lit"