CARTOGRAPHIC DISPLAY OPERATIONS ON

SURFACE DATA COLLECTED I N AN

IRREGULAR STRUCTIJRE

Douglas Cochrane

B.Sc., Simon F rase r Un ive r s i t y , 1973

A THESIS SUBMITTED I N PAXTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF ARTS

i n the Department

0 f

Geography

0 DOUGLAS COCHRANE 1974

SIMON FRASER UNIVERSITY

J u l y 1974

A l l r i g h t s r e se rved , This t h e s i s may n o t be

reproduced i n whole o r i n p a r t , by photocopy

o r o t h e r means, without permission of t h e au tho r .

APPROVAL

Name :

Degree :

T i t l e of Thesis :

Douglas Cochrane

Master of A r t s

Cartographic Display Operations on Surface Data Col lec ted i n an I r r e g u l a r S t r u c t u r e

Examining Committee:

Chairman: M.C. Kellman

Thomas K. Peucker - Senior Supervisor

-

Robert B. ~ o r s f a g

Theodor D. S t e r l i n g

Car l %arm Ass i s t an t o fe s so r

Univers i ty of Washington S e a t t l e

PARTIAL COPYRTClIT LICENSE

I hereby g r a n t t o Simon F rase r Univers i ty t he r i g h t t o lend

my t h e s i s o r d i s s e r t a t i o n ( t h e t i t l e of which i s shown below) t o u se r s

of t he Simon F r a s e r Un ive r s i t y Library , and t o make p a r t i a l a r s i n g l e

copies only f o r such u s e r s o r in response t o a reques t from the l i b r a r y

of any o t h e r u n i v e r s i t y , o r o ther educa t iona l i n s t i t u t i o n , on i t s 'own

behal f o r f o r one of i t s u s e r s . I f u r t h e r agree t h a t permission f o r

mu l t ip l e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted

by me or t he Dean of Graduate S tudies . It is understood t h a t copying

o r pub l i ca t ion of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l no t be allowed

without my w r i t t e n permission.

T i t l e of Thesis / ~ i s s e r t a t i o n :

Cartographic display operations on surface data collected i n an

irregular structure.

Douglas Cochrane

(name )

November 1, 1974

( d a t e )

ABSTRACT

Cartography is the art of displaying scaled reproductions of geographic

areas or spatial phenomena. The procedure, or methodology, by which

cartography is performed may be broken down into three basic steps;

data collection, data storage and data display. Each step involves

the manipulation of information; information degcribing that which

the cartographer has attemped to reproduce. Information must flow

through the process from the encoding of the original geographic area

or spatial phenomena to the cartographic display. The process in its

totality must be capable of maximum information retention, and, since

the process is only as strong as its weakest step, each step must in

turn be capable of maximum information retention,

The research involved in this thesis has been carried out for the

purposes of maximizing both the information capacity and the efficiency

of the cartographic process for computer cartography. A relatively

new geographic data structure has been designed for the storage of

information representative of surfaces. Specific points chosen for

their ability to characterize both changes and similarities of a surface

are collected and stored in triangular networks. The author's major

contribution is a system of computer programs for the displayof the

new data structure. Three basic routines were developed; a contour

plotting routine, a perspective view routine and a routine for

construction of shaded relief maps. The objective of each routine was

the display of all information contained by the data structure, and for

a properly designed data structure this should be all the information

collected from the surface. Sample runs of the system have been made

and one such run is included accompanied by the programs and subroutines

of the system. Sample displays of the test surface seem to imply that

the data structure and display routines do indeed retain the majority

of the information collected.

(iii)

ACKNOWLEDGEMENTS

I wish t o thank my sen io r supe rv i so r , D r . T.K.Peucker, f o r suggest ing

the top ic of t h i s t h e s i s and f o r h i s a s s i s t a n c e i n t h e r e sea rch , A g r e a t

d e a l of thanks goes a l s o t o D r . R.B.Horsfal1 f o r h i s encouragement

throughout t h e r e sea rch and f o r h i s welcome comments i n proof reading

t h e d r a f t . A s p e c i a l thanks must go t o my fe l low graduate s tuden t

M r . David H. Douglas no t only f o r t h e va r ious subrout ines and programs

used by t h i s system bu t a l s o f o r h i s encouragement and many h e l p f u l

comments throughout t h e e n t i r e p r o j e c t .

F i n a l l y I wish t o express my most s i n c e r e thanks t o my wi fe , Vickeri

E. Cochrane, f o r h e r encouragement, understanding and pa t ience .

TABLE OF CONTENTS

ABSTRACT

ACKNOWLEDGEMENTS

TABLE OF CONTENTS

TABLE O F FIGURES

CHAPTER I

CHAPTER I1

CHAPTER 111

CHAPTER I V

C W T E R V

APPENDIX I

APPENDIX I1

THE CARTOGRAPHIC PROCESS

THE COLLECTION STAGE

THE STORAGE STAGE

THE DISPLAY STAGE

C o n t o u r ing

B l o c k D i a g r a m s

Analytical H i l l S h a d i n g

CONCLUSION

PROGRAMS AND SUBROUTINES

A SAMPLE DATA STRUCTURE

SELECTED BIBLIOGRAPHY

page

iii

iv

v

v i

1

9

19

2 8

3 1

3 5

4 7

5 4

56

85

8 8

TABLE OF FIGURES

Figure Page

Car tographic Pyramid 2

Cartographic Pyramid 8

Regular Grid Data Col lec t ion 11

Line Reduction 13

Autocorrelat ion Function 14

L i s t Processing Records 2 1

Data Flow, L i s t Processing 2 2

Tr iangula t ion Scheme, Canoe River Valley 2 4

I n i t i a l Po in te r Choice 2 5

Canoe River Valley, Or ig ina l Topographic Map 29

Canoe River Valley, 100 Meter Contours 3 2

Canoe River Valley, 50 Meter Contours 33 \

Canoe River Valley, 100 Meter Contours 3 4

Canoe River Valley, Viewpoint within t h e Data S t ruc tu re 36

Canoe River Valley, 200 Unit P r o f i l e Spacing 3 7

Canoe River Valley, 500 Unit P r o f i l e Spacing 3 8

Rotat ion of Data S t ruc tu re 3 9

P r o f i l i n g of Data S t ruc tu re 4 0

Horizon and P r o f i l e Compar'ison 4 1

Canoe River Valley, O r thographic P ro jec t ion 42

Canoe River Valley, True Perspec t ive P ro jec t ion 4 3

Canoe River Valley, Orthographic P ro jec t ion 44

Canoe River Valley, True Perspec t ive P ro jec t ion 45

Canoe River Valley, O r thographic P ro jec t ion 4 6

Vectors of a Typical Tr iangle 4 9

Shading Symbol Range 50

Canoe River Valley, Shaded Relief Map 5 3

Figure

V e r t i c a l Plane T h r o u g h the D a t a S t ructure

ISOKONT

EYEBALL, READER1

GREY SHDE

GREY SHDE

T R I G L E

SCHNET

SCHNET

SCHNET

S CHNET

SCHNET

C O W E R

CONVER

C O W E R

ANGFND

MARK I T

I S O N , LOR and LAND

WORK1, WORK2 and F I N D I T

SORTER and S T R I P

ROTATE and CENTRE

MAXMIN and P O I N T S

OUTS I D

WORK1 and BORDER

I r r egu la r D a t a S t ruc ture

R e c o r d s of the D a t a Structure

CHAPTER I : THE CARTOGRAPHIC PROCESS

Cartography is the technique of collecting and displaying information

pertaining to the spatial characteristics of geographic regions.

Collection is the gathering of information, while display involves

the reproduction of this information in such a way as to give the user

or viewer maximum knowledge of the original region or spatial phenomena.

Since geographic study areas are in general very large and the amount of

collected information quite voluminous, certain storage techniques must

be used to keep the information in a logical, usable form. Storage,

the intermediate step to collection and display, should be based on

consideration of the information being stored and the use which is to

be made of the information.

However, cartography is not simply a matter of collecting, storing and

displaying information - cartography is the totality of these processes and must consider each one as part of the group. Common to each of

these stages is that which is of primary concern; the information!

Since the objective is to reach display with as little loss of inform-

ation as possible and since the efficacy of each step is dependent

upon that of the previous step, it is necessary that retention be

maximized throughout. The maximum amount (in terms of desired output)

of relevant information must be collected, since the base of the cart-

ographic pyramid is the collection stage (Figure 1.1) and all else

is dependent upon this. The cartographic pyramid gives a good illustr-

ation of information flow. The shape of the pyramid illustrates inform-

ation retention; information content generally decreases with height,

or the processing distance, from the original region.

p hlc

Flow

Or ig ina l Geographic Region

Figure 1.1 Cartographic Pyramid

Reg lo n

The information content of cartographic maps has recently come

under study by both cartographers and mathematicians. Sukhov ( 26 ) , examines how the contents of a map may be generalized and, through

the use of information theory, calculates the deviation of a given

map from that of a fully "loaded" source, or what proportion of the

maximum possible information is not on the finished map. Gokhman

and ~ekler ( 12 analyse the methodology in studying the information

content and capacity of maps, deriving various formulas which may be

of great use in the ~tudy of maps as the end product of the mapping

process. These studies, which consider the map as an information

source, are of great value in the optimization of the cartographic

technique - since a good measure of the efficiency of the technique is in the analysis of the information content of the finished product

(the map).

In all studies of information flow and the communication of ideas

it is necessary to understand how information is measured. How much

information does any given communication have or not have? If a map

contains information is this information in the best form, is the

map a good medium for the transportation of information, and, how

may the map be improved to maximize its information content with

little or no loss of aesthetic value?

Information theory is that body of descriptive knowledge built up

to describe and quantify the content of communications. A certain

communication is deemed to carry information only if it alleviates

or reduces the uncertainty which existed for the receiver before that

particular communication took place. The true quantity of vagueness

removed is difficult, if not impossible to measure. However through

statistical treatment it has been possible to assign relative values

to the information content and to evaluate certain modes of communic-

ation on this basis, In other words it is impossible to assign an

absolute value to the amount of information contained in a given

message, however tkcontent can be compared to other messages and

other modes of communication for a relative measure.

Communications may be said to relate information to the observer

concerning an event (E) which occurs with a probability of P(E).

If the observer is told that the event (E) has occurred then by

definition he has received

units of information.

The base chosen for the logarithm dictates the units of information.

The logarithm to base 10

gives information in units of Hartleys. With the natural logarithm

or log t o base e

information i s i n n a t s . The most common, i n computer usage, i s t h e

base 2 logari thm o r b ina ry u n i t i n which t h e r e s u l t i n g u n i t of

information

i s c a l l e d the - b i t .

For example i f t h e event (E) i s t h e f l i p p i n g of a co in and t h e

subsequent observa t ion of a head then P(E) = $5. I f t h e co in i s

f l i pped and a head i s observed then t h e informat ion rece ived i s r i

= log2 P(E) = log2 (2) = 1 b i t lLJ 1 ) = 0 [$J = log lo (2) = 0.301 Har t leys .

An information-generating mechanism is an informat ion source i n .

which t h e success ive ly emi t ted symbols a r e s t a t i s t i c a l l y independent

of each o the r and i s c a l l e d a zero-memory information source. Each

symbol from t h e f i x e d f i n i t e source a lphabet

a r r i v e s from t h e mechanism wi th no dependence whatsoever on the pre-

ceding symbol. The source (S) i s completely descr ibed by the a lphabet

and p r o b a b i l i t i e s wi th which t h e symbols occur - which a r e

Since this is a discrete information source, with all symbols being

statistically independant, the information provided by the occurrence

of the symbol S is given by j

The average amount of information per symbol can then be calculated as

-P(s~) . I(Si) bits i =O

Average information per symbol is defined as the entropy of the source

and may be interpreted as the amount of uncertainty of the observer

before the communication is received.

Information theory sets out the guidelines for the quantification of

the information contained in a given conmunication, through the appl-

ication of which, various modes of communication may be compared for

the most efficient. Abramson ( 1, p.13) presents a comparison of the

spoken word of a radio announcer to that of one single television

picture. Assuming that the radio announcer has a vocabulary of 10,000

words, of which he selects 1000 in a random fashion (which is of course

not the case in the process of normal speech), the probability of him

choosing any one particular 1000 sequence (E) is then

And the amount of information transmitted by any particular sequence (E)

will be

= 13,000 bits

The television picture is composed of a matrix 500 x 600 with each

cell of the matrix (black and white television) having any one of

10 distinguishable grey scalee possible. Therefore the base alphabet

for the television is

Thus the probability of any one picture (E) occurring is

1

And the information transmitted by one of these pictures is

Therefore, comparatively speaking, the information transmitted by one

television picture far exceeds the information transmitted by 1000

spoken words .

Maps, as Sukhov (26) , and Gokhman and Mekler ( 12 ) point out,

may also be compared in this manner. Hqever, since maps are a

combination of various communication techniques, { i . e. text, topographic line work (isolines, hachuring), thematic line work

(roads, settlements), hydrographic line work (rivers, lakes, oceans),

colour patterns (tonings, shadings), and map graticules each of

which may be considered as an information-generating mechanism,

information content must be determined for each of these techniques

considered as an information source. The map must be broken down

into its component parts and analysed piece by piece, the information

for the various sources is then summed to give the total.

However the effects of various tehniques working in conjunction with

each other will be lost with this method. If for example five hundred

foot colour changes are overlaid with ten foot contours of a topographic

surface it is possible to calculate the information transmitted by the

contours and by the colour changes, however the information of the

combination is difficult if not impossible to measure. It is known

that the colours will enhance the contour levels and make the relief

appear more definite - but the information transmitted by this tech- nique is not a hard and fast unit which can be measured. The map

therefore contains certain tangibles which may be quantified in terms

of the information they provide, but the intangibles are inherent in

the map and any information which they provide will be overlooked in

objective measurements even though it may be of great value.

Analysing the information content of given display techniques will

afford some insight into the efficiency of the displays, however

within the cartographic process, display is simply the end result.

If different maps vary in information it cannot be concluded that

this has been solely determined by the display technique. The cart-

ographic pyramid, which illustrates information flow and whose shape

corresponds with information retention, shows that information is not

only lost at the display stage but also at the collection and storage

stages. A good display can do no more than to illustrate the inform-

ation that has proceeded this far in the process; it cannot generate

its own information but must be fed this from the lower stages. Thus

analysis of information content must take place at the collection,

storage and display stages in order to optimize the cartographic process.

The collection stage should give the base information set for the study

and should be based upon that which the study is intended to examine. The

collection stage is therefore the most important member of the group

and all necessary information should enter here. Entrance at this point

should be directly into storage, with the storage technique used being

developed primarily to serve the information entering. However, since

the information must: exit from the storage stage to the display stage

the storage technique should also accommodate the exiting of the inform-

ation. Thus the storage technique assumes a dual role - it must not only

accept information from t h e c o l l e c t i o n s t a g e wi th maximum r e t e n t i o n

but a l s o g ive information t o t h e d i sp l ay s t a g e , i d e a l l y , e x a c t l y a s

much information a s was accepted from the c o l l e c t i o n s t age . I f t h e

s to rage and d i s p l a y s t a g e s can be c rea t ed t o c a r r y a l l information

c o l l e c t e d , t h e ca r tog raph ic pyramid w i l l assume a new shape (Figure

1 .2) - s i n c e a l l information c o l l e c t e d from the o r i g i n a l geographic

reg ion w i l l appear on t h e reduced car tographic vers ion . However

s i n c e the car tographer does not wish t o d u p l i c a t e r e a l i t y - only

accentua te c e r t a i n chosen a t t r i b u t e s - the c o l l e c t i o n s t a g e w i l l - - ---__

O r i g i n a l R e g i o n

Figure 1 .2 Cartographic Pyramid wi th information r e t e n t i o n .

s t i l l show some l o s s of information. This cannot r e a l l y be termed

a l o s s , a s i t i s a f i l t e r i n g o r s e l e c t i v e sampling, record ing only

those elements of r e a l i t y which a r e deemed necessary t o an understanding

of t h a t p a r t i c u l a r phenomena which is t o be s tudied .

CHAPTER 11: THE COLLECTION STAGE

Maps are two dimensional representatiors of three dimensional geog-

raphic areas or spatial phenomena. They are based on information

selected from an infinite number of characteristics concerning the

area in question. Since the map only represents reality in a reduced

form it is imperative that the information selected from reality be

characteristic of the reality. However, since no map is a represent-

ation of all of reality but only certain portions thereof, the

information selected must only characterize these portions. For

example, topographic maps, hydrographic maps, land use maps, settlement

pattern maps or linguistic maps all represent different portions of

reality and the information selected from reality must characterize

the appropriate portion. Therefore, the type of map to be constructed

or the focus of the information to be presented by the map, dictates

what information shall be collected from the geographic area.

In the representation of surfaces (topographic, economic, or social),

specific, characteristic points must be selected from the real surface.

These points are functions of two variables

in which f is the function of surface behaviour from which z may be

calculated - given x, y and the function. However, unlike mathematical

surfaces, f is generally unknown for topographic surfaces. Therefore

in the collection of information from a topographic surface it is

necessary to collect the three values, x, y, and z , which will completely

define the behaviour of the surface. Selection of these points may be

carried out by any one of four methods;

1) regular grid overlay in the x, y plane, with collection of the

2; variable,

2) sampling varying x, y coordinates for a fixed z value with constant

incremental increases in z (i.e. recording x, y coordinates of isolines),

3 ) collecting z values for random x, y coordinate locations,

or 4) collecting z values of specific points on the surface - where the points are determined by how characteristic they are of changes in

the surface.

Any one of these methods will yield data characteristic of the surface

and enable reproduction. However, since some collect more redundant

information than do others, they require larger samples to achieve

equality of information content.

Sampling a surface with the regular grid overlay technique involves

superimposing a grid of constant x, y increments onto the surface and

recording the z values for the grid nodes, In order that a statistically

sound sampling be obtained, the theoretically "best" cell size may be

calculated with the sampling theorem, which states that "if a function

has no spectral components of frequency higher than W, then the value

of the function is completely determined by a knowledge of its values

placed $ W apart. The lack of freedom of the function for variation

from the prescribed path between sample points is a consequence of its

lack of spectral components of frequency higher than W". (28). For

example, if undulations of the surface are to be detected to within

100 units, it would be necessary that the grid cells be of 50 unit

spacing or less.

Data collection based on regular spacing of the sample points should,

if the sampling theorem is invoked, yield a statistically sound

representation of the surface. Data collected in this manner tend to

have the appearance of overlaying the surface with a "fish net",

Figure 2.1, since each cell of the grid is only represented by the

grid nodes.

Figure 2.1 Regular Grid Data Collection. Program: VUBLOK; Production: D. Cochrane

Information collected in this manner is somewhat random, since the

collection points (nodes of data collection) are not chosen for the

surface in question but for the reproduction desired by the user. It

is unlikely, unless the cell size chosen is extremely small, that the

nodes will happen to correspond with breaks bf the surface, or points

which define changes in the surface. If the cell size is made small

enough to ensure collecting all important points of the surface, the

amount of data thus collected will be enormous, the majority being

redundant. For example, if a data collection area is comprised of

mountainous regions and plateau regions interspersed, the cell size

will have to be extremely small to sufficiently cover the mountainous

regions ensuring adequate data collection points. However, the plat-

eaus will be represented by more than an adequate number of data points,

each representing nearly the same surface function and many of which

may be considered redundant. Collection of data by this method not

only wastes a good deal of time in the collection stage, but since the

data is carried through and displayed, waste is perpetuated throughout

the process.

Sampling a surface by recording only the x, y coordinates of isolines,

or, allowing the x, y values to vary while holding the z value fixed is

a technique of prime importance to photogrameters. Through the use of

optics, the photographer constructs, from aerial photographs, a three

dimensional visual representation of the surface, from which he then

records the x, y coordinates of the surface by a form of tracing.

Sampling a surface by this technique involves recording the x, y coord-

inates of major breaks along contour lines, the interval at which data

points are recorded is dependant upon the regularity of the surface.

Smooth contours may be approximated by few data points, while rough and

irregular contours will need more. However, as pointed out by Douglas

and Peucker ( 11 ), attempts to automate this technique have lead to

vast amounts of redundant information being recorded, since the machine

does not intuitively "choose" points which represent major breaks of the

surface. Automatic line followers record data at regular intervals

of either distance or time, and the sampling is therefore random

since points are not chosen to represent the surface, only the output

desired. Of course, the theoretically "best" interval may be chosen

with the sampling theorem but this does not alleviate the problem of

vast amounts of redundant information being reporded.

Redundant information may be screened from the raw data. Douglas

and Peucker develop a technique which seems to relieve the data of

only its redundant parta. Figure 2.2 shows the comparison between

raw data and screened data - line a is constructed from 112 data

points, while line b is constructed from only 19 data points.

Figure 2.2 Line Reduction. Douglas and Peucker (11)

Information content of the message (line) has been retained, only the

redundancy has been removed. Of course line a or line h is only a

generalized representation of the real line since specific breaks have

not necessarily been recorded. However, this technique of choosing a

vast number of data points and subsequently screening out only the

redundant information will approximate methods of recording only points

of high information content. As the number of data points chosen increases,

so does the probability of choosing points of high information content - i

points representing changes in the surface function.

A surface is a continuum of points each of which is characteristic

of its neighbourhood. A neighbourhood may be defined mathematically

as "for any real number c, zt neighbourhood of c is that open interval

containing cl'. ( 14, p. 74 ) The size of the interval of which the

data point is characteristic is of course dependant on the nature of

the surface or the surface function, For example, the neighbourhood

of which a point on an infinite level plane is characteristic is

an infinite neighbourhood, whereas if the surface is extremely

irregular the neighbourhood of characterisation becomes smaller, W'

The size of the neighbourhood will vary in relation to the irregularity

of the surface in a given distance. The autocorrelation function

shows that the larger a neighbourhood becomes, the more probable it

is that any point chosen at random from the real surface will vary

from the data point for which the neighbourhood is being drawn. J'

C o r r e l a t i o n C o r r e l a t i o n

Distance

Figure 2.3 Autocorrelation Function

Figure 2.3 illustrates the autocorrelation function - correlation of the surface at any given data point is plotted against distance

from the data point in question. The correlation between the surface

and the data point decreases as the distance increases, maximum

correlation is attained at the data point where the distance is

equal to zero. 1

Neighbourhoods, or regians of homogeneous information, are comprised

of characteristically similar data but similar data carries redundant

information, unnecessary to the description of the surface. Changes

in neighbourhood however are of high information content, as in the

statement "the wrong side of the tracks"; the tracks are an important

socio-economic boundary relaying a great deal of information to the

observer. A line or boundary has a certain extension or neighbourhood

within which a point may lie and still be descriptive of the change

between neighbourhoods. If a point is chosen at random from a surface

then the probability that this point will lie within a neighbourhood

is proportional to the area of the neighbourhood, and the probability

that the point will lie on a neighbourhaodboundary is proportional / to the "area" of the boundary. Thus if the probability of choosing a

point within a neighbourhood (WN) is equal to P(WN) and the probability

of choosing a point on a neighbourhood boundary (NB) is equal to P ( N B )

then

since the area (WN) will generally exceed the "area" (NB). The inform-

ation content of the points so chosen is

from 1

and

which implies d i r e c t l y t h a t

The information provided by a po in t chosen from wi th in a neighbourhood

w i l l be less than t h a t provided by a po in t chosen on a neighbourhood

boundary - assuming of course t h a t 1 i s t rue . Thus, i f t h e neighbour-

hood of a p o i n t i s known, and i t should be, simply from t h e r e g u l a r i t y

of t he s u r f a c e , sampling of t he s u r f a c e f o r t he r e t e n t i o n of high

information con ten t need only cons ider t h e c h a r a c t e r i s t i c s of p o i n t s

which r ep re sen t changes of neighbourhood. This of course is t h e method

of t he surveyor who records only breaks of t h e s u r f a c e , which he has

apprsximzted with pclygccs (n~ighhiirhoods).

P /

Po in t s c h a r a c t e r i s t i c of changes i n neighbourhoods have been c a l l e d

"sur face s p e c i f i c " p o i n t s by Warntz ( 2 9 ) . H e o u t l i n e s t h r ee d i s t i n c t

groupings of p o i n t s and l i n e s ; peaks and p i t s , passes and p a l e s , r i d g e

l i n e s and course l i n e s . With t h i s s e t a l l high l e v e l changes i n su r f ace

behaviour (neighbourhood changes) may be noted. A peak is a po in t of

l o c a l maximum and a p i t a p o i n t of l o c a l minimum ; both a r e , w i th in

some r e l a t i v e neighbourhood, surrounded by c losed contours . A pass

and a pa l e a r e p o i n t s of i n f l e c t i o n i n t h e s u r f a c e p r o f i l e and have no

c losed contours i n t h e i r neighbourhoods. A pass occurs a t t h e j unc t ion

between r i d g e l i n e s and course l i n e s , while a pa l e occurs a t t h e h igh

po in t between p i t s , and i s i t s e l f a low po in t of t h e r i d g e between t h e

p i t s . A r i d g e i s the l i n e of meeting of two upwardly s lop ing s u r f a c e s

and conversely a course is a l i n e of meeting of two downwardly s lop ing

surfaces. Ridge l i n e s , which connect points of l o c a l maxima, a r e

separated by course l i n e s , which connect po in t s of l o c a l minima.

Surface s p e c i f i c po in t s a r e po in t s of high information content , not

only a s changes i n neighbourhoods but a l s o a s s p e c i f i c po in t s wi th in

groupings of neighbourhoods. The knowledge t h a t a point is a peak

conta ins the information t h a t t h i s point is a l o c a l maximum, t h e f i r s t

de r iva t ive of the su r face funct ion is zero a t t h i s point , and t h i s

point represents a change of surface c h a r a c t e r i s t i c s f o r a t l e a s t two -I

neighbourhoods (usual ly more). En t i r e groups of neighbourhoods can be (*

reconstructed from the su r face s p e c i f i c points , s ince not only do

these po in t s des ignate changes i n neighbourhoods, they - a r e c h a r a c t e r i s t i c \ . ~J

of the neighbourhoods of which they form a p a r t . These po in t s de f ine

the s p a t i a l r e l a t ionships of neighbourhoods through t h e i r r e l a t i v e

pos i t ions on the su r face (21, p.29).

Sampling only su r face s p e c i f i c po in t s i s not a new technique - surveyors

have been co l l ec t ing su r face d e t a i l i n t h i s manner f o r cen tu r i e s .

However, wi th in t h e f i e l d of computer cartography, even though da ta

have been co l l ec ted i n t h i s manner i t is normally in terpola ted t o a 1 I

regular g r i d f o r s torage . Symap, one of the forerunners i n computer '

cartographic processing, accepts a s input i r r e g u l a r l y co l l ec ted da ta \ i po in t s but transforms the da ta onto a regular g r i d f o r any subsequent \ opera t ions . I n t e r p o l a t i o n d i spe r ses t h e information content of a

I i

da ta point to the g r i d nodes. Information regarding the t rue , surface

re l a t ionsh ip of a point , which is c h a r a c t e r i s t i c of a neighbourhood,

to other neighbouring s p e c i f i c po in t s is l o s t . ..-...-..- A l l information which

is i m p l i c i t t o a p o i n t ' s pos i t ion , and, a l l information t o be derived

from a p o i n t ' s i n t e r a c t i o n wi th i ts neighbouring s p e c i f i c po in t s - a l s o impl ic i t by p o s i t i o n - is gone. Cer ta in ly a new type of impl ic i t -

ness is gained, t h a t of the pos i t ion of the g r i d nodes, but t h i s i s

f o r the in te rpo la ted surface , not the t r u e geographic surface f o r

which the da ta was co l l ec ted . Surveyors, on the o ther hand, u t i l i z e

t h i s information. Once da ta concerning t h e surface s p e c i f i c p o i n t s has

been c o l l e c t e d t h e surveyor p l o t s the p o i n t s and connects them (even i f

only menta l ly) i n t o a rough t r i a n g u l a r network. From t h i s network he

then begins t o draw h i s conclusions about t h e su r f ace , whether i t be

f o r subd iv i s ions , engineer ing s t r u c t u r e s ( roads , b r idges , b u i l d i n g s ) ,

o r simply t o map the s u r f a c e - h i s dec i s ions a r e always based on t h e

s u r f a c e s p e c i f i c p o i n t s and t h e i r neighbourhoods. Once such d a t a has

been c o l l e c t e d i t i s imperat ive t h a t every d e t a i l be r e t a i n e d ;

in format ion content should not s u f f e r simply t o add t o t h e ease of

c o l l e c t i o n o r handling.

THE STORAGE STAGE

Computer representation and manipulation of surfaces involves

processing vast amounts of data, which, if left in a raw un-

structured form would be generally inaccessible. Data sets are

comprised of "elementary data items" ( 10, p.118 ) which are

singular pieces of information (i.e,x-coordinate of data point).

A collection of all elementary data items concerning one data

point is termed a record, and the collection of records into

logical units is a file. "The arrangement and interrelation of - records in a file form a data structure" ( 1 0 1 1 ) . The

information to be retrieved and the type of processing to be

carried out dictates the type of data structure best suited to

a given problem. As Dodd (10) points out there are many types

of data structures and many combinations thereof.

As an example of a data structuring problem, consider all the

names and phone numbers of the greater Vancouver region recorded

as one name, address and phone number (elementary data items) per

data card (record). If these cards were then shuffled randomly

they would constitute a very raw data set, and, there would exist

many possible structurings.

a). alphabetize the records by surname,

b). alphabetize the records by given names,

c). sequence the records by phone numbers,

d). regionalize on the basis of city blocks and order by

house number,

or e). sort by street then order by house number.

Any of these methods would yield a data structure, however the speed

of computer access to the elementary data items would vary for each.

The alphabetized structure of (a) would be very efficient for

retrieval of phone numbers if given the surname, but, the structure

of (e) would only impede the same search.

The functions of a data structure are numerous and depend entirely

on the problem to be solved. Surface storage renuires data

structures which are capable of a). minimizing space requirements

b). allowing fast access for retrieval,

updating, and storage

and c). retention of maximum spatial

information content.

However, simply attaining each of these does not guarantee a good

data structure - these characteristics must compensate and cornpl.rment each other, attaining the best from each in the totality.

As mentioned previously, retention of maximum spatial. information

content is of prime importance, si.nce this is the purpose of the

data structure in surface manipulation. There would be absolutely

no benefit in creating a fast, small data structure containing

relatively little information.

Surface representation is best accomplished by the collection of

irregular data points, surface specific points. These points carry

information describing the spatial relationships of certain surface

features and this information must be retained by the data structure.

Any data structure designed to accommodate this information must \ have the following characteristics.

Neighbourhood relationships should remain with the data, to the point

of making them explicit by actually recording within the data structure

the fact that two or more certain neighbours are adjacent and have a

given relationship between them. Pointers between neighbours such as,

point A is a neighbour of point B which is in turn a neighbour of point

C, should be established. The surface specific points should he related

to each other within the structure, since a good deal of information

is herein contained.

Linked or - list organization is a type of data structure in which

the links "are used to divorce the logical organization from the

physical organization" ( 10, p.122 ) of the structure. The links,

or as they are more generally called pointers, connect logically

adjacent parts of the data set which are stored in physically non-

adjacent areas of storage, thus allowing the computer to process a

record which is in a physically different area of storage than that

adjacent to the preceding record. The pointers between records may

be part of the actual record or they may simply be computer addresses

calculated at the time of processing (hash coding technique). For

example in Figure 3.1 if the data were to be organized alphabetically

it would be structured and processed as it is in the table.

Label Name

Anderson, Q. J.

Bambi, D.R.

Doe, J.

Smith, E.J.

Vanline, W.K.

Address

1715 W. Hastings St.

307 Cordova St.

1219 E. South St.

2102 S. West St.

1605 N. Smith St.

Phone #

397-0421

472-9152

281-0415

972-0916

104-3721

Pointer

Figure 3.1 List Processing Records

But if it was desired to process the names sequentially by phone number

from lowest to highest it would not be necessary to even check the

phone numbers, processing would start at Vanline and proceed as shown

in Figure 3.2

Figure 3 . 2 Data Flow, L i s t Processing

Vanl ine ' s p o i n t e r is 12 which means t h e next record t o be processed 1

has a l a b e l of 12 and i s Doe, which has a p o i n t e r of 21 which p o i n t s

t o Anderson, e t c . This example demonstrates how t h e l i s t o rgan iza t ion

func t ions , wi th one l a b e l and one p o i n t e r per record . The phone

numbers become l o g i c a l l y ad jacent (process ing w i l l be i n t h e l o g i c a l

o r d e r ) even though t h e r e may be a v a s t phys i ca l d i s t a n c e between the

r eco rds .

1 l a b e l = key

A record may of course be l o g i c a l l y ad j acen t t o more than one

o t h e r record , s i n c e t he adjacency premise may d i f f e r - i . e . phone

number, s t r e e t addrese, or C h r i s t i a n name, For example t h e p o i n t e r s

e s t a b l i s h e d i n , F i g u r e 3.1 have an adjacency premise of phone numbers,

t h e phone numbers i nc rease a s t h e p o i n t e r l is t inc reases . However,

i f t h e premise were t o be surnames, t h e o r d e r of p rocess ing would have

t o be

which would i n d i c a t e a d i f f e r e n t p o i n t e r f o r each record. Obviously,

i f t h e premise were addresses another group of p o i n t e r s would be c r ea t ed .

Thus each record may have more than one p o i n t e r , with each p o i n t e r

being dependent upon a d i f f e r e n t adjacency premise. In a r e c o r d ' s

p o i n t e r l i s t t h e f i r s t p o i n t e r may po in t t o t h e ad jacent record based

on phone numbers, t h e second t o t h e ad j acen t based on street address

and t h e t h i r d based on C h r i s t i a n name. Of course each record may have

only one l a b e l s i r r e t h e record is processed by i t s l a b e l and mu l t i p l e

l a b e l s would only i n c r e a s e process ing t i m e i n checking t h e l a b e l l i s t

f o r t h e given l a b e l . Data, r ep re sen t ing s u r f a c e s t r u c t u r e s , w i l l have

m u l t i p l e p o i n t e r s , each s u r f a c e s p e c i f i c po in t must be connected t o

more than one o t h e r s u r f a c e s p e c i f i c po in t . A su r f ace cannot be

represen ted by p o i n t s which a r e func t ions o f , o r connected t o , s i n g l e

o t h e r p o i n t s .

The d a t a s t r u c t u r e chosen i n t h i s r e sea rch was one i n which m u l t i p l e d~ - -

p o i n t e r s were used t o connect s u r f a c e s p e c i f i c p o i n t s . The method

of t h e surveyor was followed; d a t a were c o l l e c t e d i r r e g u l a r l y f o r t h e

breaks of t h e s u r f a c e , o r t h e boundaries of neighbourhoods (Appendix

11, page 85 ) . The i r r e g u l a r d a t a were then t r i a n g u l a t e d wi th t h e

c r i t e r i a f o r t r i a n g l e formation being t h a t t h e p lane of t h e t r i a n g l e

most c l o s e l y approximate t h e r e a l su r f ace . The c r i t e r i o n f o r c lo senes s

was t h a t t h e t r i a n g l e chosen (many t r i a n g u l a t i o n schemes a r e p o s s i b l e

from a given number of p o i n t s ( 2 3 ,Chapter 4 ) ) should have the

l e a s t a b s o l u t e d i s t a n c e between the p lane of t h e t r i a n g l e and the r e a l

Figure 3.3 Triangulation Scheme, Canoe River Valley

su r f ace . Since the da t a poin ts f o r t h j s .structurcl were der ived

from a map i t was re ln t - tve ly simple t o c l~oosc thc plan<% o f l e a s t 2

d e v i a t i o n from t h e su r f ace . Figure 3.3 i l l u s t r a t e s t h e t r i a n ~ u l a t c d

d a t a p o i n t s ove r l a id on the o r i g i n a l sur face .

The d a t a s t r u c t u r e i s constructed from t h e t r i a n g u l a t e d network

a t t h e t ime of recording t h e da t a values. Recording i s done

according t o t h e fol lowing r u l e s :

1) s e q u e n t i a l l y ( s t a r t i n g a t 1) l a b e l a l l d a t a po in t s (nodes of

t h e network) up t o N, t h e number of po in ts i n the d a t a s e t .

The fo l lowing a r e done f o r each d a t a po in t i n o rde r , one a t a t ime

(Format i s ou t l i ned i n Appendix 11).

2 ) record t h e l a b e l (from above) and t h e x, y , z coordina te of t h e

d a t a p o i n t i n ques t ion . -----

Figure 3 . 4 I n i t i a l Po in t e r Choice

2 An automated t r i a n g u l a t i o n procedure i s being developed.

3) s t a r t i n g a t t h e p o i n t e r c l o s e s t t o no r th and proceeding i n a clock-

wise d i r e c t i o n (Figure 3 . 4 ) record t h e l a b e l s of t h e p o i n t s t o which t h e

p o i n t i n ques t ion is connected.

4 ) i f a border (ou t s ide ex t remi ty of t h e da t a s e t ) is encountered

between two p o i n t e r s , record a va lue of 32000 i n t h e p o i n t e r l i s t .

5) i f t h e number of p o i n t e r s r a d i a t i n g from the p o i n t i n ques t ion

is g r e a t e r than seven ( >7 ), then make t h e seventh p o i n t e r i n t h e

l i s t a nega t ive i n t e g e r .

The a b s o l u t e va lue of t h e i n t e g e r s h a l l depend upon, i ) t h e number

of d a t a p o i n t s i n t h e s e t (N) and i i ) t h e number of t imes , f o r

prev ious r eco rds , t h e number seven has been surpassed. I f , f o r t h e

d a t a s e t i n ques t ion , t h i s i s t h e 1st node f o r which t h e number seven

has been surpassed then t h e i n t e g e r s h a l l have a va lue of N + 1. I f ,

f o r t h e d a t a s e t i n ques t ion , t h i s i s t h e j t h node f o r which t h e

number seven has been surpassed then t h e i n t e g e r s h a l l have an a b s o l u t e

va lue of N + j .

I f t h e number of p o i n t e r s does n o t exceed seven then r e t u r n t o number

2 ) , o therwise cont inue .

8) t h e (N + j ) t h record must now be f i l l e d i n . Give i t a l a b e l of

(N + j ) and record e x a c t l y t h e same x, y , z coord ina tes a s t h a t of

t h e node i n 2 ) .

9) cont inue record ing t h e p o i n t e r s as above wi th t h e f i r s t p o i n t e r

of t h e (N + j ) t h record being t h e seventh p o i n t e r clockwise around

the po in t i n ques t ion .

10) If t h e number of p o i n t e r s happens t o exceed t h i r t e e n use the same

method a s above t o record them wi th the l a b e l of t h e new record being

(N + j + 1) o r (N + 2) .

11) hlen all pointers of the point in question are exhausted (for

the number of pointers greater than seven) make the last pointer in

the list a negative integer. The absolute value of this integer

shall be the label of the point in question (that recorded in number

2 ) ) .

Continue this process for all N data points, with the resultant (N + k) records being the data structure representing the surface, where k

is the number of nodes with more than seven neighbours.

The data structure so constructed will represent the surface not only ,,/ relative to an absolute origin, but will also contain the relative

relationships between neighbours. Processing the records of this

structure'will be carried out between logical records and not

necessarily between physically adjacent records. All the neighbours

of one point may easily be recognized by the explicit relationships

designated by the pointers, which in itseif is not an advantage

over the implicit relationship between grid nodes. However, the nodes

are surface specific points rather than simple grid node samplings

which do not necessarily characterize the surface.

CHAPTER IV: THE DISPLAY STAGE

Information extraction is the prime objective of all data collections.

Data display, for the purpose of extracting information, may be

carried out in any one of the following forms;

1) tabulate the observed data,

2) statistically analyze the data and tabulate the results,

3) graphically represent the observed data,

or 4) graphically represent the statistically analyzed data.

In all forms of display the main concern should be to maximise the

amount of information obtainable from the results (table, graph, map,

etc.) of the study. The purpose of any study and the predetermined

use to which information gathered will be applied will generally

dictate which display method is best for the data in question.

Repre~entation of a surface is dependant upon the total data, inform-

ation being drawn from the relative positions of data points. Therefore,

it is imperative that the data be observed as a whole. The parts of

the data structure have no absolute meaning and can only be judged by

comparison to their neighbours. For example, it means nothing to say

a certain data point has an x, y, z value of (100, 200, 300). The

point could be a peak of great relative significance or a point on

a relatively flat surface. Surface data should be presented to the L'

observer in a manner such that all neighbourhood relations may be easily

observed.

Cartographers, through centuries of experimentation, have developed

numerous methods for the graphical representation of surfaces. In

1701 Edmund Halley (in27) published the first isoline

Figure 4 . 1 Canoe River Valley, Original Topographic Map

mnp, of isogones o r l i n e s of equa l magnetic d e c l i n a t i o n . By 1972

when Norman Thrower ( 2 7 ) published h i s book "Maps & Man" he presented

fo r ty - th ree types of isograms a s being a s e l e c t i o n "from a much l a r g e r

number of terms f o r s p e c i a l forms of t h e isogram" ( 2 7 , p 164 ).

During the n ine t een th cen tury block diagrams o r t h r e e q u a r t e r view

p r o j e c t i o n s were developed and used p r imar i l y f o r i l l u s t r a t i n g sub-

s u r f a c e s t r u c t u r e s of t h e e a r t h i n r e l a t i o n t o t h e su r f ace func t ions .

Senefelder ( 2 7 ) , i n t h e l a t e e igh t een th cen tury , developed a technique

of using mu l t i p l e co lours o r t o n a l ranges f o r t he i l l u s t r a t i o n of

topographic r e l i e f . Hachuring and r e l i e f shading of t h e s u r f a c e was

expanded by Tanaka (27) i n t h e e a r l y twen t i e th cen tury and more

r e c e n t l y automated by Yoel i (33). H i s t o r i c a l l y , ca r tographers have

always s t r i v e d f o r t h e "best" g r aph ica l r e p r e s e n t a t i o n of t h e d a t a wi th

which they have been presen ted .

The f o c a l po in t of t h i s r e sea rch has been t h e r e p r e s e n t a t i o n of s u r f a c e s ;

sampling, s t o r age and d i s p l a y . I n o rde r t h a t t h e d a t a s t o r e d i n t he

i r r e g u l a r d a t a s t r u c t u r e be accu ra t e ly por t rayed i t was necessary t o

c r e a t e a new system of d i sp l ay r o u t i n e s t h a t were cen te red around t h e J

d a t a s t r u c t u r e . Three d i s p l a y techniques were chosen; contour l i n e s ,

block diagrams and r e l i e f shading. One system of subrout ines was then

w r i t t e n f o r each of t h e s e techniques wi th t h e r e s u l t a n t programs being

a s fo l lows; ISOKONT, a program f o r t h e product ion of contour maps,

EYEBALL, f o r t he product ion of block diagrams o r , t h r e e dimensional

views, and GREYSHDE, f o r the product ion of shaded r e l i e f maps. Using

t h e i r r e g u l a r d a t a s t r u c t u r e and r e l y i n g on numerous subrout ines

(Appendix I) these r o u t i n e s w i l l i d e a l l y accomplish t he des i r ed r e s u l t

of d i s p l a y i n g t h a t information which i s c o l l e c t e d and s t o r e d wi th a s

l i t t l e informat ion l o s s as poss ib l e .

Contour i n &

ISOKONT is a program f o r d i sp l ay ing i s o l i n e s of e l e v a t i o n (contours )

from t h e d a t a of the i r r e g u l a r s t r u c t u r e . The bas ic i d e a of ISOKONT

i s ve ry simply a s fo l lows;

Knowing t h e z va lue of t h e h i g h e s t d a t a po in t i n t h e e n t i r e

s t r u c t u r e , c a l c u l a t e f o r t h e user-defined contour i n t e r v a l

t h e va lue of t h e maximum contour l e v e l ( i . e . i f maximum z

= 510 and contour i n t e r v a l = 50, then maximum contour l e v e l

equa l s 500).

Pass t o SCHNET t h e maximum contour l e v e l and t h e va lue of t he

h i g h e s t d a t a p o i n t . SCHNET w i l l t hen , a s descr ibed i n

Appendix I, c a l c u l a t e and pl.ot a l l t h e i n t e r s e c t i o n p o i n t s

of t h i s contour l e v e l w i t h t h e edges of t h e d a t a s t r u c t u r e .

Check t o see i f a l l i n t e r s e c t i o n s of t h e p a r t i c u l a r contour

w i t h t h e edges have been found. This check i s accomplished

by s e t t i n g up a b i t o r b i n a r y ma t r ix i n which each e n t r y i n

t h e ma t r ix (each c e l l ) i s simply a binary f i g u r e ' 1 ' o r '3'.

The c e l l l o c a t i o n i s de f ined by t h e nodes a t t he end of each

edge. For example, t h e edge formed by t h e two nodes 56 and

101 w i l l have a m a t r i x l o c a t i o n of row 56 by column 101.

Each t i m e a new contour l e v e l i s s t a r t e d t h e e n t i r e mat r ix

i s set t o ' 0 ' . When an edge is passed through by a p a r t i c u l a r

contour , t h a t c e l l l o c a t i o n de f ined by t h e nodes of t h e edge

i s set t o 'I1. Thus, i n o r d e r t o ensure t h a t a p a r t i c u l a r

edge has been c rossed by t h e contour , simply check t h e b i t

l o c a t i o n .

Decrease t h e va lue of t h e contour l e v e l by t h e va lue of t h e

contour i n t e r v a l . I f any edges a r e above t h e contour l e v e l

they may be e l imina t ed from any f u r t h e r checks s i n c e t h e r e i s

no p o s s i b i l i t y of i n t e r s e c t i o n wi th the contour .

Figure 4 .2 Canoe River Val ley , 100 Meter Contours. Program: ISOKONT

Figure 4.3 Canoe River Valley, 50 Meter Contours. Program: Isokont

Figure 4.4 Canoe River Valley, 100 Meter Contours. Program: J S X O N T

. 5 ) Pass t o SCHNET t h e va lue of t he contour l e v e l and any d a t a

p o i n t which i s above t h i s l e v e l , but has not been e l imina ted

i n S tep 4) . SCHNET w i l l aga in c a l c u l a t e and p l o t t h e i n t e r -

s e c t i o n s of t h i s contour l e v e l with t he edges of t h e d a t a s e t .

6) Continue s t e p s 3 ) , 4 ) , and 5) u n t i l t h e minimum va lue of t h e

d a t a set is g r e a t e r than t h e va lue of t h e contour l e v e l .

A l i s t i n g of ISOKONT may be seen i n Figure 1.2. ISOKONT w i l l produce

maps of any d e s i r e d i n t e r v a l . Figure 4.2 and Figure 4.3 show contour

maps of t h e Canoe Val ley ( o r i g i n a l topographic map, F igure 4.1).

F igure 4.2 shows contours p l o t t e d wi th a one hundred meter contour

i n t e r v a l whi le F igure 4.3 i s of a f i f t y meter i n t e r v a l . F igure 4.4

i l l u s t r a t e s t he op t ion of having t h e contour i n t e r v a l s p l o t t e d on t h e

contour map. The d a t a set f o r t he se p a r t i c u l a r maps c o n t a i n only e i g h t y

one p o i n t s - s e l e c t e d by examination of t h e s u r f a c e , and choosing only

s u r f a c e s p e c i f i c p o i n t s .

Block Diagrams

EYEBALL i s a r o u t i n e which c a l c u l a t e s and p l o t s va r ious pe r spec t ive

view diagrams of t h e i r r e g u l a r d a t a s t r u c t u r e . For a user-def ined

viewpoint (po in t i n space from which t h e u s e r wishes t o view t h e

s u r f a c e ) and c e n t r a l p o i n t ( cen t e r of view the u s e r wishes i n t h e d a t a

s t r u c t u r e ) t h e r o u t i n e w i l l c o n s t r u c t , according t o t h e op t ion s p e c i f i e d ;

t r u e pe r spec t ive p r o j e c t i o n s , pe r spec t ive i n x and or thographic i n y ,

pe r spec t ive i n y and or thographic i n x , o r t r u e or thographic p r o j e c t i o n s .

Any viewpoint above t h e minimum z va lue of t h e su r f ace may be s p e c i f i e d

and i n f a c t t h e viewpoint may a c t u a l l y rest on t h e s u r f a c e , a l lowing t h e

u s e r t o "walk" over t he s u r f a c e (Figure 4 .5) . Also, s i n c e t h e p ro j ec t ed

s u r f a c e i s composed of s u r f a c e p r o f i l e s taken a t s p e c i f i e d r e g u l a r i n t e r v a l s

from t h e viewpoint , t h e u s e r i s a f forded f l e x i b i l i t y i n determining t h e

e x t e n t of d e t a i l on t he f i n i s h e d p r o j e c t i o n (compare F igure 4.6 t o F igure

I'ih:uri. 4 . 5 Carloc' River V a l l e y , Viewpoint With in t h e Data S t r u c t u r e . P r o f i l i n g b e g i n s a t one h a l f t h e d i s t a n c e be tween t h e v i ewpo in t and t h e c e n t r a l p o i n t . Program: EYEBALL

Figure 4.6 Canoe R i v e r Val ley, 200 U n i t P r o f i l e Spac ing . Program: EYEBALL

Figure 4.7 Canoe River Valley, 500 Unit Profile Spacing. Program: EYEBALL

A perspec t ive view of a su r f ace must show only those a r e a s of t h e su r f ace

which would be seen by t h e eye. It is t h e r e f o r e necessary f o r any prc-

j e c t i o n r o u t i n e t o e l imina te those po r t ions of t h e s u r f a c e obscured by

s u r f a c e elements i n c l o s e t proximity t o t h e viewpoint. A su r f ace po in t

m y be def ined a s hidden i f and only i f a l i n e segment connecting the su r f ace poin t and the viewpoint i n t e r s e c t s t h e s u r f a c e a t some po in t .

Kubert, Szabo, and G i u l i e r i (17) o u t l i n e a n a lgor i thm f o r t h e cons t ruc t ion

of t h i s l i n e segment f o r each su r f ace element and t h e subsequent test f o r

i n t e r s e c t i o n s . EYEBALL e l imina te s hidden s u r f a c e elements by moving a

v i s i b i l i t y p r o f i l e through t h e s u r f a c e a t r i g h t ang le s t o t h e l i n e of

s i g h t and away from t h e viewpoint.

The main r o u t i n e of t h e EYEBALL system i s simply a program t o s t a r t t h e

ope ra t ion and t o s t o p it: CONVER. The EYEBALL system ope ra t e s on t h e

i r r e g u l a r d a t a s t r u c t u r e and performs t h e ope ra t ions a s fol lows:

1) The e n t i r e d a t a s e t i s f i r s t assigned new x, y coord ina tes by

transforming and r o t a t i n g the d a t a such t h a t t h e viewpoint

Sezones t h e new x, y o r i g i n and the c e n t r a i po in t i i e s on che

x a x i s , F igure 4 .8 . The z va lues (he ight ) a r e l e f t unchanged.

Any coord ina tes now r e f e r r e d t o w i l l be t he new coord ina tes .

i V' Figure 4.8 Rota t ion of Data S t r u c t u r e

2 ) Beginning a t e i t h e r ,

P) tile ~iiit~h~lum x vaiue, o r

i i ) one half t h e d ie tance from the viewpoint to t he c e n t r a l

po in t , whichever is g r e a t e r , cons t ruc t p r o f i l e s through the

d a t a s e t , a t r i g h t ang le s t o t h e x a x i s and of a user-defined

d i s t a n c e a p a r t .

These p r o f i l e s , Figure 4.9, a r e cons t ruc ted by f i r s t c a l c u l a t i n g

t h e endpoints , which a r e simply t h e i n t e r s e c t i o n p o i n t s of t h e

boundary of t h e da t a set w i t h a l i n e of given x p o s i t i o n and

i n f i n i t e y length , po in t s A and B of F igu re 4 . 9 . The coordin-

a t e s of t h e endpoints of t h e d e s i r e d p r o f i l e a r e then passed t o

SCHNET, which determines t h e i n t e r s e c t i o n p o i n t s of t h i s l i n e

w i t h t h e d a t a s e t and r e t u r n s t h e x, y, and z coo rd ina t e s of

t h e s e i n t e r s e c t i o n s . --

Figure 4.9 P r o f i l i n g of Data S t r u c t u r e

3 ) The p r o f i l e s o r e then pro jec ted onto a n imaginary p lane which

is cons t ruc ted between t h e c e n t r a l p o i n t and t h e viewpoint.

4 ) The p r o f i l e c l o s e s t t o t he viewpoint i s then c a l l e d the horizon

and i s t e s t e d a g a i n s t t h e second c l o s e s t p r o f i l e t o t h e view-

po in t t o s e e i f t h e second c l o s e s t p r o f i l e i s v i s i b l e . The

h ighes t vfs ibPe p o i n t s (of t h e horizon and t h e p r o f i l e ) then

become t h e new horizon, F igure 4.10. Th i s new horizon is then

compared t o t h e next p r o f i l e away from t h e viewpoint and a new

horizon can be formed. Each time a comparison is done t h e

rwwltant tuwiwoa Le ploc lgd , w h @ t h s ~ t t $a wtrtr@ly new ~f

j u s t the same a s t h e preceding horizon. This process cont inues

u n t i l t h e l a s t p r o f i l e , which is less than (by l e s s than one c e l l

width) o r equal t o t h e maximum x va lue of t h e d a t a set, has been

compared t o t h e next t o last horizon and has been p l o t t e d .

A l i s t i n g of t h e program EYEBALL may be seen i n F igure 1.3.

- Hovlxon

---- Prof i le

. N e w H o r l r o n

,-, H l d d e n S o c t l o n r

Figure 4.10 Horizon and P r o f i l e Comparison

Some of t h e p l o t s which r e s u l t from var ioue viewpoints being used wi th

EYEBALL a r e a s fol lows;

Above each p l o t i t is noted on o contour map the d i r e c t i o n from which

the v i e w was taken.

Fi gurc. 4.11 Canoe River Valley, Orthographic Pro j ect ion. Program: EYEBALL

F i g u r e 4 . 1 2 Canoe R i v e r V a l l e y , True P e r s p e c t i v e P r o j e c t i o n . Program: EYEBALL

Figu re 4.13 Canoe River Val ley , Or thographic P r o j e c t i o n . Program: EYEBALL

Figure 4.14 Canoe River Valley, True Perspective Projection. This view is of the north west half of the data structure only. Program: EYEBALL

I Viewing

I Direct ion v

Figure 4.15 Canoe River Valley, Orthographic Projection. Program: EYEBALL

Figure Number Descr ip t ion

4.11 Orthographic view of t h e su r f ace wi th 100 u n i t spacing

of t h e p r o f i l e s .

True pe r spec t ive p r o j e c t i o n wi th 100 u n i t spacing of

p r o f i l e s .

Orthographic view wi th 100 u n i t spacing of p r o f i l e s .

True pe r spec t ive p r o j e c t i o n of only ha l f t h e d a t a

s t r u c t u r e (nor th west corner only) w i t h 100 u n i t

spacing of p r o f i l e s .

4.15 Orthographic view wi th 100 u n i t spacing of p r o f i l e s .

Ana ly t i ca l H i l l Shading

The GREYSHDE system i s used f o r t h e c o n s t r u c t i o n of shaded r e l i e f models

of t h e s u r f a c e a s developed by Yoel i (31) . The method i s somewhat

s i m i l a r t o t h a t of Yoel i except t h a t Yoel i c a l c u l a t e s t h e shading of

u n i t squa re s of t h e d a t a set and GREYSHDE c a l c u l a t e s shading f o r e n t i r e

t r i a n g u l a r polygons of t h e d a t a s t r u c t u r e . The i r r e g u l a r d a t a s t r u c t u r e

being comprised of t r i a n g u l a r polygons would make i t imprac t i ca l t o

r e d i v i d e t h e s u r f a c e i n t o smaller r e g u l a r polygons s i n c e t h e r e s u l t a n t

g r i d c e l l s would be of t h e same s u r f a c e f u n c t i o n a s t h e l a r g e r t r i a n g l e s .

Thus t h e on ly p r a c t i c a l method by which t o shade t h e s u r f a c e is us ing

t r i a n g l e s .

A s Yoel i p o i n t s o u t i t is impossible t o c r e a t e a n exac t r e p r e s e n t a t i o n

of t h e e a r t h ' s r e l i e f on a map. However by i l l umina t ing a s u r f a c e w i th

a p o i n t l i g h t source and reproducing t h e shading which appears on t h e

su r f ace , a c l o s e approximation t o r e a l i t y may be r e a l i z e d . Yoel i beg ins

h i s d e r i v a t i o n of t h e t h e o r e t i c a l shading of a su r f ace by using Wiechel 's

equat ion f o r l i g h t i n t e n s i t y a t any po in t on a sur face .

cos (e ) = c o s (a) . c o s (b) + s i n ( a ) . s i n (b) . c o s (c )

In which:

e is the ang le between the l i g h t r a y and t h e normal t o t h e su r f ace

a t t h e po in t i n ques t ion .

b t h e s lope ang le of t h e su r f ace .

a t he complement of t h e ang le of s l o p e of t h e l i g h t r a y ,

c t h e ang le between t h e s u r f a c e and a p lane perpendicular t o t h e

l i g h t r ay .

Yoel i u ses vec tor a n a l y s i s t o apply t h e equat ion t o an a r e a element

deiined by two v e c t o r s i n t h e plane of i l l umina t ion . The v e c t o r s a r e

Then, assuming t h a t t h e l eng th of t h e l i g h t vec to r 8 ( l i g h t r ay ) is one

u n i t t h e equat ion

Sx(a b - a b ) + S (a b - axbZ) + SZ(a b - a b ) Y Z Z Y Y z x X Y Y X

cos (e) =

- a b ) 2 + (azbx - axbz)2 + (a b - a b )2 d ( a y b z . x Y Y x

is der ived . Th i s i s a s i m p l i f i c a t i o n of ~ i e c h e l ' s formula f o r t h e l i g h t

i n t e n s i t y of a r ec t angu la r s u r f a c e element.

A s mentioned above, i t would be imprac t i ca l t o d i v i d e t h e t r i a n g u l a r

polygons i n t o r ec t ang le s o r squares . Each t r i a n g l e of t h e i r r e g u l a r

d a t a s e t is defined by t h r e e edges o r vec to r s which a r e i n t he p lane

of t h e polygon they de f ine , thus any two edges of any p a r t i c u l a r 3 -+

t r i a n g l e can be thought of as the v e c t o r s a and b.

Given t h e coord ina tes of p o i n t s L, M and N i n F igure 4.16 a s ,

it i s a simple ma t t e r t o c a l c u l a t e t h e components of t h e v e c t o r s -b -t a and b.

I f t h e assumption is then made t h a t t h e l i g h t source i s i n t h e 0

northwest and a t an ang le of i n c l i n a t i o n of 45 t h e components

of t he vec tor may be e a s i l y found. Having found t h e va lues of

S S S *---"---- -.3 "L -+ -b

L V ~ C L W C L W L L L ~ the cclu~ponentcl uf a aid 3 , cos e m y be x' y' z solved f o r .

L

Figure 4.16 Vectors of a Typical T r i ang le

'Cos e ' is the va lue of t he l i g h t i n t e n s i t y over the given s u r f a c e

e l -men t - The l i g h t i n t e n s i t y over t h e s u r f a c e v a r i e s from whi te

wi th f u l l i l l umina t ion t o b lack wi th no i l luminat ion . F u l l

i l l umina t ion r e s u l t s when t h e ang le of i l luminat ion corresponds t o

t h e s u r f a c e normal wi th l i g h t r a y s h i t t i n g t h e sur face perpendicular ly

y i e ld ing a v a l u e of 1 .00 f o r cos e . When t h e sur face normal corresponds

t o t h e a n g l e of inc idence of t h e l i g h t r ay and is i n t h e same d i r e c t i o n

a s the l i g h t ray , t h e v a l u e of c o s e i s -1.00; however t h i s r a r e l y occurs

i n n a t u r e ( i . e . a n overhanging c l i f f , o r double valued s u r f a c e func t ion ) .

Surface s lopes f o r topographic s u r f a c e s a r e r a r e l y over fo r ty - f ive degrees,

which inc l ined away from t h e l i g h t r ay w i l l g ive cos e a va lue of 0.00.

Since f o r s lopes of over f o r t y - f i v e degrees away from the l i g h t ray , t h e

s u r f a c e would appear very heav i ly i n shade, a l l values of c o s e l e s s than

o r equal t o 0.00 may be considered a s black. Thus t h e range of cos e

from -1.00 t o 1.00 w i l l correspond t o a s u r f a c e shading range of b lack

to whi te wi th a l l c o s e v a l u e s l e s s than o r equal to zero having a b lack

shading. For t h e purposes of GREYSHDE t h i s range was d iv ided i n t o t e n

groups and ass igned shading symbols from a l i g h t grey t o black, F igure 4.17.

0 .1 . 2 .3 .4 .s .6 .7 . a .9 1.0

( I - cos e)

Figure 4.17 Shading Symbol Range

These symbols when used r e l a t i v e t o each o ther g ive a good impression

of surface relief and i r r e g u l a r i t y .

The a lgor i thm f o r GREYSHDE i s a s fol lows;

1) Read i n i r r e g u l a r da t a s t r u c t u r e .

2 ) C a l l TRIGLE, a subrout ine which forms t h e v e r t i c e s of a l l 1

t r i a n g l e s i n t o t h r e e one dimensional a r r a y s . For example,

i f t h e j t h t r i a n g l e encountered has t h e ve r t ex numbers of

12, 13, 14 then a r r a y one con ta ins 1 2 i n t he j t h p o s i t i o n ,

a r r a y two con ta ins 13 i n t h e j t h p o s i t i o n and a r r a y t h r e e

con ta ins 14 i n t h e j t h pos i t i on .

3 ) C a l l SHADE, a subrout ine which c a l c u l a t e s the grey shading

of every t r i a n g l e i n t h e d a t a s t r u c t u r e by t h e method

ou t l i ned above.

4) Using SCHNET c a l c u l a t e t h e x, y coord ina tes of t h e i n t e r s e c t i o n

p o i n t s of t he da t a s t r u c t u r e and a l i n e of cons tan t y va lue

passed through the d a t a s t r u c t u r e from minimum x t o maximum x.

The s t a r t i n g , o r i n i t i a l , y va lue of t h e l i n e is t h e maximum

y va lue of t h e da t a .

5 ) Moving from t h e minimum x va lue of t h e i n t e r s e c t i o n p o i n t s

found, t o t h e m a x i m u m , p a i r t h e i n t e r s e c t i o n p o i n t s a s fol lows;

( l s t , 2nd), (2nd, 3 r d ) , (3rd, 4 t h ) , . . . . . . . . , ( (n - l ) s t , n t h ) .

6 ) Calcu la t e t h e x coord ina te of t h e midpoint of each p a i r and

determine w i t h i n which t r i a n g l e of t h e da t a s t r u c t u r e each

midpoint l i e s .

7) Assign t h e shading va lues of t h e app ropr i a t e t r i a n g l e s t o t h e

s e c t i o n s of t h e l i n e represented by each i n t e r s e c t i o n p a i r .

Therefore, from the f i r s t i n t e r s e c t i o n po in t t o t h e second

i n t e r s e c t i o n po in t t h e l i n e w i l l be assigned a shading va lue

of t he t r i a n g l e wi th in which the l i n e between t h e i n t e r s e c t i o n

p o i n t s l i e s .

8) Assemble and output a l i n e of p r i n t ( fo r t h e l i n e p r i n t e r ) ,

each cha rac t e r of which i s based upon t h e shading va lue i t i s

t o r ep re sen t , Figure 4.17.

9) Decrease t h e y va lue of t h e l i n e ( a s ca l cu la t ed i n 4) by a

c e r t a i n i n t e r v a l and c a l c u l a t e t h e i n t e r s e c t i o n p o i n t s of

t h i s l i n e wi th the d a t a s t r u c t u r e .

10) Go back t o s t e p 5) and cont inue t h i s process u n t i l t he minimum

y va lue of t h e da t a s t r u c t u r e i s encountered by t h e l i n e moving

downward through the s t r u c t u r e .

A l i s t i n g of t he GREYSHDE program may be seen i n F igure 1 .4 . The product

of GREYSHDE is a shaded r e l i e f map (Figure 4.18) which is a d i s p l a y of a

much more genera l n a t u r e than e i t h e r t h e contour map o r t h e pe r spec t ive

view. However, the shaded r e l i e f map does c a r r y information of a much

higher l e v e l and c o n s t i t u t e s g e n e r a l i t i e s r a t h e r than p a r t i c u l a r s .

F i g u r e 4.18 Canoe R i v e r V a l l e y , Shaded Relief Map. Program: GREYSHDE

CHAPTER V: CONCLUSION

Nearly every field of academic work has felt the impact of the

computer. With its extensive storage facilities, rapid access to

data and high speed of calculation the computer has the possihility

of revolutionizing cartography. As a communicative process, the

object of cartography is to convey to an independant observer as

much information concerning a geographic area or spatial phenomena

as is possible. Information is generally lost at each of three stages

within the process; collection, storage and display. This thesis

has been an endeavor to illustrate how, with an alternative approach

to data collection, and through implementation of the addressing

facilities offered by the computer, retention of information can

be maximized throughout the process. The approach recommended as

an alternative to reguiar grid collection and/or storage is that of

collection of surface specific points and storage of the absolute

coordinates of these points and their spatial relationships to

neighbours with a linked list data structure.

One important innovation of the data structure developed is its

ability to convey and retain information about the relation between

surface specific points. This is important because a surface function

is dependant upon the relation and interaction between points for

the majority of surface information. A regular grid will of course

yield this information, however the amount of necessary surface data

must be increased along with the processing time. The linked list

data structure appears, by the tests run thus far, to maximize inform-

ation retention while saving on data storage space and processing time.

The systems for the dfsplay of the irregular data structure have been

designed to give the user flexibility in viewing the surface. From

t h e f i l t e r e d o r smoothed r e p r e s e n t a t i o n a f fo rded by GREYSHDE t o t h e

minute i s o l i n e d e t a i l of ISOKONT, the use r may choose the amount of

information t o be displayed. GREYSHDE, through use of t h e t r i a n g l e ,

o r neighbourhood, a s t he lowest l e v e l d a t a i tem, f i l t e r s o r channels

t h e information i n t o groups, g iv ing t h e u s e r i n s i g h t t o t h e r e l a t i o n -

s h i p s of neighbourhoods. GREYSHDE g ives no a b s o l u t e s u r f a c e va lues ,

only the s u r f a c e func t ion r e l a t i o n s . EYEBALL a l lows t h e use r t o view

t h e s u r f a c e from a n o w 1 pe r spec t ive , aga in no a b s o l u t e s p e c i f i c s

of t h e s u r f a c e appear on the reproduct ion , on ly t h e r e l a t i o n s h i p s

between neighbourhoods. However, t h e s e r e l a t i o n s h i p s may be f u r t h e r

enhanced by a change of p r o f i l e d e n s i t y , a change of viewing p o s i t i o n

o r a change of p ro j ec t ion . The most d e t a i l e d d i s p l a y i s given by

ISOKONT, wi th t h e use r d i c t a t i n g t h e amount of in format ion t o be on

t h e f i n i s h e d p l o t simply through adjustment of t h e contour i n t e r v a l .

These systems a l low t h e information of t h e s u r f a c e s p e c i f i c p o i n t s

r e l a t e d through p o s i t i o n t o be p l o t t e d w i t h a h igh degree of in format ion

r e t e n t i o n . The d i s p l a y s are, a s = e a r l y a s p o s s i b l e , t h a t in format ion

which has been c o l l e c t e d , no l o s s can b e a t t r i b u t e d t o s t o r a g e o r

ou tput (d i sp l ay ) techniques.

The p r i n c i p l e s developed i n t h i s t h e s i s should s e r v e a s a b a s i s t o

ongoing r e sea rch f o r t h e development of e f f i c i e n t geographic d a t a

s t r u c t u r e s . Geography, be ing a s tudy of dynamic phenomena, needs d a t a

s t r u c t u r e s capable of handl ing eve r changing s p a t i a l d i s t r i b u t i o n s .

Data s t r u c t u r e s which a r e a b l e t o cope w i t h t h e i n t e r a c t i o n s and i n t e r -

r e l a t i o n s h i p s between groups of d a t a . The d a t a s t r u c t u r e h e r e i n

developed may not be capable of s o l v i n g t h e s e problems b u t should

h e l p i n t h e development of new s t r u c t u r e s ; i f on ly t o p o i n t ou t a

need f o r development.

APPENDIX I: PROGRAMS AND SUBROUTINES

The t h r e e d i s p l a y r o u t i n e s o u t l i n e d i n Chapter I V were dependant upon

numerous sub rou t ines f o r t h e i r e f f i c i e n t ope ra t ion . One of t h e s e was

of major importance and w i l l be d iscussed i n some d e t a i l wh i l e t h e

r e s t a r e s e l f explanatory and may be seen i n F igure 1 - 2 t h r u F igure 1.23. I

SCHNET is a r o u t i n e f o r searching and developing a pa th through t h e

i r r e g u l a r d a t a s t r u c t u r e . The pa th chosen by SCHNET i s based on s e v e r a l

p o s s i b l e c r i t e r i a , one of which must be suppl ied by t h e invoking program.

The fo l lowing c r i t e r i a were chosen f o r SCHNET;

a ) The pa th of t h e J t h i s o l i n e - f i n d t h e i n t e r s e c t i o n p o i n t s

of a pa th c r ea t ed by a p lane of he igh t J i n t e r s e c t i n g t h e

d a t a s t r u c t u r e . SCHNET w i l l r e t u r n t h e x , y coo rd ina t e s of

t h e i n t e r s e c t i o n p o i n t s .

b ) The pa th of a v e r t i c a l p l ane - f i n d t h e i n t e r s e c t i o n p o i n t s

of a v e r t i c a l p lane pass ing through t h e d a t a s t r u c t u r e ,

SCHNET w i l l r e t u r n t h e x, y , z coord ina t e s of t h e pa th .

See F igure 1.1.

c ) The d a t a p o i n t c l o s e s t t o a given x , y coord ina te . Given

any x, y coord ina te , p r e f e r a b l y w i t h i n t h e d a t a s e t boundaries ,

SCHNET w i l l r e t u r n t h e l a b e l of t h e d a t a p o i n t which i s c l o s e s t

t o t h i s p o i n t .

The pa th fo l lowing a lgo r i thm f o r a network was f i r s t developed by

Dayhoff (9) and used f o r contour ing a r e g u l a r g r i d p a t t e r n i n X-Ray

c rys t a l log raphy . The a lgor i thm, a s used by,SCHNET, i s b a s i c a l l y t h e

same a s Dayhoff 's except t h a t minor changes have been made t o accommodate

both a new d a t a s t r u c t u r e and t h e c r i t e r i a of a v e r t i c a l p lane .

B a s i c a l l y SCHNET t e s t s each edge i n a sys temat ic manner f o r an i n t e r s e c t i o n

wi th t h e pa th . A few d e f i n i t i o n s a r e necessary p r i o r t o a complete

d e s c r i p t i o n of t h e algori thm:

RP - r e f e rence po in t , normally t h e major node of t h e edge being

t e s t e d . Swinging always occurs around t h e RP.

SP - sub p o i n t , is the secondary node of t h e edge t h a t i s being

t e s t e d . Always t h e changing p o i n t , o r t h e po in t which i s

be ing changed i n t h e swinging procedure.

EP - edge p o i n t , is t h e p o i n t which i s being c a l c u l a t e d a s t h e

i n t e r s e c t i o n between t h e pa th and t h e edge def ined by t h e

nodes RP and SP.

The a lgor i thm f o r SCHNET is;

SCHNET i s passed, from t h e invoking procedure, t h e f i r s t

RF t o be used, t h e va lue of which i s dependant upon the

c r i t e r i a chosen. For t h e c r i t e r i a of fol lowing a contour

pa th , RP i s chosen a s any d a t a p o i n t which i s above t h e

va lue of t h e contour l e v e l (pa th t o be followed) bu t has

no t been e l imina ted due t o subsequent process ing ( see

write-up of ISOKONT). For t h e c r i t e r i a of fol lowing t h e

pa th of a v e r t i c a l p lane ( s t r a i g h t l i n e ) RP i s chosen a s

t h e d a t a p o i n t c l o s e s t t o one end of t he v e r t i c a l p lane

( i . e . t h e v e r t i c a l p lane becomes a l i n e i n t h e x, y p lane

def ined by two p o i n t s a t t h e ends of t h e l i n e 1 This po in t

may be found wi th a subsequent c a l l t o SCHNET us ing c r i t e r i a

c ) from above. For t h e c r i t e r i a of f i n d i n g t h e d a t a p o i n t

c l o s e s t t o a g iven x, y coo rd ina t e l o c a t i o n t h e RP used ,ay

be t h e l a b e l of any d a t a p o i n t .

2) SP i s chosen as t h e d a t a p o i n t po in ted t o by t h e f i r s t

( s e q u e n t i a l l y measured) p o i n t e r i n RP'S p o i n t e r l i s t .

3) The edge formed by RP - SP i s checked f o r an i n t e r s e c t i o n

(EP) wi th t h e pa th . )

4) a ) i f one e x i s t s i t i s c a l c u l a t e d and s t o r e d . SP i s then

s e t t o t h e d a t a va lue pointed t o by t h e next ( s e q u e n t i a l l y )

po in t e r i n RP's p o i n t e r l i s t - 3) i s repea ted un le s s of

course t h e next p o i n t e r i n RP's l i s t i s zero o r blank - i n d i c a t i n g t h a t a l l RP's neighbours have y i e lded i n t e r s e c t i o n

p o i n t s . Which f o r t h e contour pa th i n d i c a t e s a c losed

cont inuous contour around RP i n which c a s e c o n t r o l i s re turned

t o t h e invoking procedure. For t h e v e r t i c a l p lane c r i t e r i a

t h i s i s impossible un le s s of course RP is on a boundary of

t h e d a t a s e t and then one of RP's p o i n t e r s would have a va lue

of 32000 which would i n d i c a t e t o SCHEJET t h a t t h i s cond i t i on

may occur and t o check f o r i t .

b) i ) i f no EP i s found, and, no EP has y e t t o be found

f o r t h i s p a r t i c u l a r RP, then SP i s s e t t o t h e d a t a va lue

poin ted t o by t h e next ( s e q u e n t i a l l y ) p o i n t e r of RP's p o i n t e r

l i s t . 3) i s repea ted u n l e s s of course t h e next p o i n t e r i n

RP's l i s t is zero o r blank.- i n d i c a t i n g t h a t a l l RP's neighbours

have been checked and found t o y i e l d no pa th i n t e r s e c t i o n s .

Control i s r e tu rned t o t h e invoking r o u t i n e which must g ive

SCHNET a new RP.

i i ) But, i f no EP i s found and one was found f o r t h e l a s t

SP ( i . e . 4a was performed f o r t h e l a s t p o i n t e r i n RP1s l i s t )

then 6 i s performed.

6 ) RP is s e t t o t h e d a t a va lue of SP and SF i s s e t t o t h e d a t a

va lue poin ted t o by t h e p o i n t e r i n t h e new RPls l i s t immediately

fo l lowing t h e p o i n t e r t o t h e o l d RP.

7) The edge RP - SP is checked f o r an EP. One must e x i s t s i n c e

t h e pa th has en tered t h e t r i a n g l e and un le s s i t stopped i n s i d e

t h e t r i a n g l e ( t h e end of a s t r a i g h t l i n e ) i t must l eave t h e

t r i a n g l e through t h i s s i d e ( s ince i t d i d no t l eave through

t h e s i d e def ined by the nodes o l d RP, SP.

8) SP i s s e t t o t h e d a t a va lue poin ted t o by t h e next p o i n t e r i n

RP's l i s t .

9 ) Go back t o s ta tement 3 ) .

This process of swinging through RP's and SP's w i l l cont inue u n t i l one

of t h e s topping mechanisms i s encountered. The s topping mechanism f o r

a contour pa th could be one of two cond i t i ons ;

a ) t h e contour i s a continuous c losed curve which even tua l ly

returns cc the initial R?.

b ) t h e contour encounters one of t h e boundaries of t h e da ta s e t

and must s t o p , i n which case SCHNET t r a n s f e r s t h e pa th fol lowing

procedure back t o t h e beginning of t h e pa th a t t h e i n i t i a l RF.

SCHNET then cont inues t o fo l low t h e pa th i n t h e oppos i t e d i r e c t i o n

u n t i l ano the r border i s reached i n which c a s e t h e c o n t o u r has

been completed and c o n t r o l i s r e tu rned t o t h e invoking procedure.

The s topping mechanism f o r a s t r a i g h t l i n e may be one of two events

occuring

a ) a boundary of t h e d a t a s e t i s reached, b u t s i n c e t h e i n i t i a l RP

was one end of t h e l i n e o r where t h e l i n e en t e red t h e d a t a s e t

a l l i n t e r s e c t i o n s have been found and c o n t r o l i s r e tu rned t o

t h e invoking procedure.

b ) an end p o i n t of t h e l i n e h a s been reached and s i n c e t h e i n i t i a l

RP was one end of t h e l i n e o r where t h e l i n e en t e red t h e d a t a s e t

I ,

Figure I. 1 Vertical Plane Through the Data St rur- t tlrl , S(;;?J'' o ~ ~ ~ z ~ - i t ' on the data structure and the vrr t l r:i ; 4 ' : intersection points o f i'ne t - r w

c o n t r o l is re turned t o t h e invoking procedure.

A sample SCHNET run a s i l l u s t r a t e d i n F igure 1.1 i s a s fo l lows;

I n i t i a l l y RP i s 1

I

None

H

G

F

None

E

None

D

C

None

l3

A

None

The f i n a l SP being 32000 i n d i c a t e s t o SCHNET t h a t a border has been

reached. The va lues of EP as i n t e r s e c t i o n coord ina t e s a r e then

r e tu rned t o t h e invokipg procedure.

The l i s t i n g f o r SCHNET may be seen i n ~ i ~ u r e I - 7 t h r u F igure I. 11 .

T11e programs of t h e d i s p l a y system u s e t h e s u b r o u t i n e s a s f o l l o w s ;

I SOKONT

CONVER

CENTRE

ROTATE

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SORTER

T h e s u b r o u t i n e SCHNET always u s e s t h e f o l l o w i n g s u b r o u t i n e s ;

SCHNET

WORK1

BORDER

PLOTS

FACTOR^ PLOT^ NUMBER^

LAND

LOR

ANGFND

MARKIT

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4

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h

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D

l ST

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(NP

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(:!R

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2)

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(Y,)

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(f.R

)-(D

EL

l*(Z

(NR

)-Z

(!IP

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L

GO

T

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9

C

IF

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E

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ER

LAP

S

OR

Ll E

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F T

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TH

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CO

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l LU

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20

4

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IRIT

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O

TO

2

08

2

11

Y

bI(

K)=

Y(N

9)

Zti

(K)-

Z(t

lR)

K=

K+

l Y

lI(K

)=Y

(PiP

) Z

N(K

)=Z

(NP

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Fig

ure

1.8

SC

HN

ET

C

on

tin

ue

d

I:iF

LAG

-.T

RU

E.

20

8

K=

K+

i 2

13

K

L-1

IT

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1P=I

SO

::(:iR

,ItP

,KL)

IF

(t

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GO

TO

2

b5

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FIN

D O

UT

IF O

!iE

OF

TH

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PIN

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LIN

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RD

ER

IF

(t:P

T(J

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Q.3

20

00

) G

O

TO

20

3

IF(t

iPT

(J+

WL

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4E

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00

0)

GO

TO

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05

C

FIN

D O

UT

IF

EIT

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E

ND

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OR

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CH

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ET:

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20

3

XS

=X

2+

10

0.

YS

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2

26

75

C

AL

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CR

OS

S(X

(~:R

),Y

(~~

R),

X(N

P),

Y(N

P),

X~

,Y~

,X~

,YS

,A,B

,~~

~X

) IF

(tll

X.!

!E.C

) G

O

TO

23

65

X

5=

X2

Y

5='1

2+13

0.

GO

TO

2

07

5

20

65

IF

(tlI

X.E

Q.2

) G

O

TO

99'3

C

IF T

HE

E

t:3P

OIt

ITS

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F

TH

E

LIf

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H

AP

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N T

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CO

INC

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OR

I

F O

NE

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F

TH

E

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ON

T

HE

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R

LIN

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N

ITS

EN

DP

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T

HE

N

C0N

TIt

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2

50

Y

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CS(A

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PS)G

O TO

2

52

z:

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(NP

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S(B

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S)

GO

TO

2

60

i IF

(AB

S(A

-Xl)

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.EP

S)

GO

T

O

26

01

f!I

XO

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RljE

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(CA

L)

GO

TO

2

60

IF

(K.G

T.2

) G

O

10

26

0

CA

LL

AIIG

F::O

(Xl,X

2,Y

l,Y2,

X,

Y,fIP

T,M

,NR

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f!P=I

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J,

ti

n)

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1

0 G

Y

IF(A

BS

(B-Y

21

.GT

.EP

S)

GO

TO

2

b0

IF

(l<

SS

(:%

-X2)

.GT

.EP

S)

GO

TO

2

63

G

O

TO

25

11

IF

(AU

S(R

-Yl)

.GT

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S)

GO

TO

2

>3

IF

(AB

S(A

-Xl)

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EP

S)G

O

TO

25

3

f!l X

O=.

TRU

E.

Dl b

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X(I

!Pl-

X(N

It1

U

IbIi

=Y

(!!?

)-Y

(NR

) DEL=SQRT(DISTl*DlSTl+DlST2+DIST2)

Dl S

Tl=

A-%

(:JR

) D

l ST

Z-C

-Y(I

!R)

DELl=S~RT(UlfTl*DISTl+DlST2*DIST2)

Z>

I(K

)=Z

(!iR

)-(D

EL

l+(Z

(t.I

R)-

Z(i

jP))

)/D

EL

CO

TO

2

63

.;

(AB

S(B

-Y2)

.GT

.EP

S)C

O

TO

25

4

IF(A

BS

01

-XZ

).G

T.E

PS

)GO

TO

2

54

N

I XO

=.F

ALS

E.

DIS

Tl=

X(t

iP)-

X(N

R)

Dl S

TZ

=Y

(:JP

)-Y

(f:R

) DEL=SQRT(DISTl+DISTl+DIST2rDIST2)

Dl S

Tl=

A-X

(t;R

) D

l ST

2=

B-Y

(NR

) DEL1=SQRT(DISTl*OISTl+DIST2*DIST2)

Z!l

(K)=

Z(

t!R

)-(O

EL

l*(Z

(RR

)-Z

(W)

))/D

EL

G

O

TO

26

0

CO

t:T I Ii

UE

K

L=

1

I TE

IIP

=I S

Ot!(

NR

,I.IP

,KL)

K

=K

+1

IH

FLA

G-.

TR

UE

. G

O

TO

3

C K

OV

E

TH

E

PC

lNT

ER

OV

ER

O

NE

P

OS

lTlO

tl

IN N

R'S

L

IST

, W

HE

RE

T

HE

DIR

EC

TIO

N

C D

EP

EN

DS

O

N

TH

E

VA

LUE

O

F

MU

L f1

.E.

OR -

2 O

TF

LAG

-.

TRU

E.

Fig

ure

1.9

SCHNET

Co

nti

nu

ed

IF(:

i3C

R.E

Q.0

) G

O

TO

4

93

35

CC

1.T

I !:U

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(I';

iLA

S.A

:dO

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FLA

G)

GO

TO

4

':!,:l

=o

IF(J

.LE

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CA

LL

F

IfiD

IT(J

,NR

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N,N

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T)

GO

TO

7

4 IF

(K.G

E.2

00

) G

O

TO

10

C

IF

TH

E

FO

LL0L

II:i

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T:,O

IF

ST

AT

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EN

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A

RE

T

RU

E

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HE

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A

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CTED

T

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T.T

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GO

T

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69

10

T

ES

T=.

FA

LSE

. LL

AB

=.F

ALS

E.

!14=

LA2

!IR=!

:P

CO

TO

6

!!o=:

:R

:i?=:

!P

GO

TO

G

x!:

(~)=

xx

(i)

Tll,?

L'-

.FA

LSE

. Y

I;(K

)=Y

:J(

1)

IF(T

HR

U)

K=

K-1

IF

(!;R

IT.?

;E.O

) G

O

TO

99

9

CA

LL

f:t

,RK

I T

(XN

, YI

I,K,

LS

I G,X

FAC

T,C

ON

T,

LE

T)

IF(K

.LT

.23

0)

GO

TO

9

99

K=l

IF(K

.CT

.1)

GO

T

O

15

G

O

TO

95

9

99

3

IF(K

.LT

.2)

GO

T

O

99

9

C

IF

A

BO

RD

ER

H

AS

B

EE

N

RE

AC

HE

D

TH

EN

ilE

MU

ST

S

AV

E T

HE

V

ALU

ES

O

F A

LL

C

IN

TE

RS

EC

TI0

I;S

A

ND

G

O

BA

CK

T

O

TH

E

NR

A

T W

HIC

H

WE

EN

TE

RE

D T

HE

D

AT

A

tin'=

tl

~i

T

ES

TS

. F

ALS

E.

LLA

C=

.FA

LSE

. K

=2

K

EO

R-1

C

AL

L

FI?!

DIT

(J,N

R,N

Q,N

,;(P

T)

IF(J

.GT

.1)

GO

TO

3

ti ti

>I = 0

CA

LL

FI

ND

IT(J

,I.1

R,I

lNN

,H,N

PT)

GO

TO

7

KL

=-1

K=l

IF(.

;IO

T.L

LAB

) G

O

TO

881

TE

ST

=.

Frt

LSE

. LL

AB

=.F

ALS

E.

!IQ

=LA

B

I!R=:

IP2

CA

LL

F

I:ID

lT(J

,l~

R,t

:Q,N

,NP

T)

GO

TO

7

r:p=

rin

liR

=iI

P2

SET

- -

C Q

fIC

E

A

RO

RD

ER

H

AS

B

EE

N R

EA

CH

ED

FO

R

TH

E

SEC

OI:O

T

INE

IF

1 O

NE

R

UN

WE

C

KtJO

tI T

HA

T

TH

E

PA

RT

ICU

LA

R

CO

NTO

UR

II

E H

AV

E

BE

EN

MO

RK

ItiG

O

N

HA

S

C

EX

ITE

D F

RO

f:

TH

E

DA

TA

SE

T

ArJD

LI

E C

AN

T

HE

RE

FO

RE

CO

XC

AT

EN

AT

E

Tti

E

TbiO

C

G

RO

UP

S

IN E

lTll

ER

UlR

EC

TlO

EI

FR

OM

TH

E

OR

IGIt

iAL

N

R

AN

D

PLO

T

TH

E

C R

ES

ULT

AN

T

VE

CT

OR

S

OF

V

ALU

ES

. D

O

99

5

I=l,

KM

l P

X( I N

E)=

XN

( I)

PY

(IN

E1

-YN

(I

Fig

ure

1.10

SCHNET

Continued

::::X

:;=K

-l

IF(;

!:IX

:l.L

T.l)

G

O

TO

3

I F('d

4X:I

.IdE

.l)

GO TO 1000

IF(iiRIT.NE.1)

GO T

O 1

000

t1P=

NP2

GO T

O 69

1000

R

ETU

RN

EN

D

Fig

ure

1-11

SC-H

NET

Continued

CA

LL

14A

X!.I

IN(X

,XB

(2),

N,

J.N

I4N

)

TH

E

PU

RP

OS

E

OF

T

iiIS

S

UB

RO

UT

INE

IS

TO

C

ALC

ULA

TE

A:

LD

PLO

T

TH

E

HO

RIZ

ON

S O

F E

YE

5AL.

L.

AC

TU

ALL

Y

C0:

iVE

R

SIM

PL

Y

SE

TS

U

P T

HE

C

ALC

ULA

TIO

NS

, W

HIC

H

AR

E

TH

EIi

P

ER

FOR

i4E

D

Bf

VA

RIO

US

O

T!iE

R

SU

6RO

UT

INE

S.

CO

iIVE

R

IS N

EC

ES

SA

RY

S

INC

E

TH

E

DA

TA

C

OI.l

ES

IN T

O

TH

E

SYST

E!:

IN P

L-1

A:

ID

TH

E

SU

BR

OU

Tli

iiS

A

RE

A

LL

IN

FO

RT

RA

II.

PA

RA

ME

TE

RS

:

X,Y,

Z T

llE

C

OO

RD

IHA

TE

S

OF

TH

E

IRR

EG

ULA

R

DA

TA

S

TR

UC

TU

RE

S

TO

RE

D

IN N

LO

NG

A

RR

AY

S.

NP

T

TH

E

PO

IXT

ER

S O

F T

HE

D

AT

A

ST

WC

TU

RE

S

TO

RE

D

IN

A 7

aY

N

A

RR

AY

. N

T

HE

T

OT

AL

I:U

:IBER

O

F

PO

INT

S

IN

TH

E

DA

TA

S

TR

UC

TU

HE

. 3

OT

ii

RE

AL

ATJD

D

UK

MY

. N

PR

OJ

TH

E

TY

PE

O

F

PR

OJE

CT

ION

D

ES

IRE

D.

-SE

E

EY

EB

AL

L

FOR

D

ET

AIL

S.

RE

AD

(S,l

CO

)(C

PT

(I

),

1-1,

3)

RE

AD

(5,l

Ol)

C

EL

LS

1

00

~

02

14

At(

3F

lO. 0)

10

1

FO

XlA

T(F

5.

2 C

AL

L

CE

NTR

E(X

,Y,X

C,E

) C

C

TH

IS S

EC

TIO

N

SE

TS

U

P T

HE

B

OR

DE

R

00 2

2

LVP

=1.3

V

PN

(LV

P)=

VP

T(L

VP

) 22

C

PtI

(LV

P)=

CP

T(L

VP

) C

AL

L

RO

TATE

(XB

,YB

,LL,

VP

N,C

PN,

LN

)

...

-.

I PE

tI=

-3

CA

LL

P

LOT

(AX

.AY

, I P

EN

)

!!S

EC

=l

CA

LL

P

RJI

Ll(

VP

T,

CPT

, PX

, PY

, PZ

,NN

N,

NSE

C,N

NN

,NSE

C)

DM

Yl*

AB

S(P

Y(2

)-P

Y(l

))

DM

YZ

=A

BS

(PY

(S)-

PY

(3))

D

MX

l=A

BS

(PX

(2)-

PX

(l))

D

MX

2=

AB

S(P

X(4

)-P

X(3

))

FA

C=.

25

C

AL

L

FA

CT

OR

(FA

C1

LI

4=1

CA

LL

RO

TATE

(X,Y

,N,

VPT,

CPT

, L

N)

IiN

ii=O

C

AL

L I

:AX

MIN

(X,X

MIN

,N,

J,N

NN

)

Fig

ure

1

.12

C

ON

VER

A:lA

=!:A

Xl(

Dr:

Yl,

DM

Y2,

DM

Xl,

DM

X2)

F

AC

= lO

.O/A

IlA

C

AL

L

FA

CT

OR

(FA

C)

C

CR

OS

S W

ITH

C

OR

DER

IIK

PT

=XI!

I N

H

AL

FD

=C

PT

(l)/

Z.

t;co

u=o

IF(1

:KP

T.

LT

.HA

LF

0)

WK

PT

-HA

LFD

C

AL

L C

RDb~

(WKP

T,YM

AX,W

KPT8

YMIN

0XB(

1)8Y

B(1)

,XB(

2)8Y

B(2)

8XC,

YC,N

lX)

I~C

oU=t

:CoU

* 1

:iD=:

:C

K=

5 IF

(III

X.E

Q.1

) GO

TO

1

0

IF(t

IIX

.NE

.4)

GO

T

O

5 G

O

TO

1

0

K=

<+

l R

X(V

)=X

C

C'io

:) =Y

C

GO

TO

1

4

K-K

+l

nx

(K)=

xc

U

Y(K

)=Y

C

CA

LL

CRO

SS(W

KPT,

YMAX

,WKP

T,YM

~N,X

B(~)

,YB(

~),X

B(~)

,YB(

~),X

C,YC

,NIX

) IF

(:II

X.r

4E

.l)

GO

-TO

1

45

IF

(K.E

Q.0

) GO

T

O

16

GO

TO

1

75

IF

(:II

X.:

iE.C

) G

O

TO

1

7

GO

TO

17

5

K=

K+

l B

X(K

)=X

C

BY

(K

)=Y

C

IrR

I T-2

I:

CA

LL=O

N

30

R=

4

IF(K

.LE

.2)

GO

T

O

17

53

C

AL

L

ST

RIP

(BX

,BY

,NE

OR

) IF

(NK

PT

.EQ

.XM

IN)

GO

T

O

75

12

IF

(WK

PT

.EQ

.XM

AX

) G

O

TO

7

51

2

AA

=A

III I

Il(E

Y(l

),B

Y(2

))

IF(A

A.I

:E.B

Y(l

))

GO

T

O

75

11

B

Y(l

)=B

Y(l

)-E

PS

2

BY

(2)=

BY

(2)*

EP

S2

G

O

TO

7

51

2

BY

(l)=

GY

(l)+

EP

SL

B

Y(2

)=3

Y(Z

)-E

PS

2

CA

LL

SC

HN

ET(

X,Y

, Z,

NPT

,RP,

C

O,N

D,N

,X(N

D),

BX

(l),

Y(N

O),

,BY

(I),

INR

I T,!l

CA

LL,

Ll,

L2,

XFO

, L

ET

) N

CA

LL=O

N

RIT

=l

IF(A

ES(B

X(l)

-BX(

2)).

GT.€PS)

GO

T

O

75

1

IF(A

BS(B

Y(1)

-BY(

2)I.

GT.E

PS)

GO

T

O

75

1

:IN

Xtl

=l

DO

7

49

I=

l,N

IF(ARS(X(1)-GX(I)).GT.EP

S)

GO

T

O

74

9

IF(AGS(Y(1)-BY(l)).GT.EPS)

GO

T

O

74

9

CO

tlT I N

UE

X

N(l

)=B

X(l

) Y

I!(l

)=U

Y(l

) G

O

TO

9

7

CA

LL

SC

ifNET

(X,Y

,Z,N

PT.R

P,C

, L2

,N,B

X(l

),B

X(2

).B

Y

(l).

BY

(2

),N

RIT

.

i!S

EC

=l

CA

LL

P

RJ:

Jl(V

PT

,CP

T,

XI;,

YN,Z

N,

NN

XN

,NS

EC

,ND

IM,N

SE

C)

CA

LL

S

TR

I P

(XN

,YN

,NN

XN

) C

E=.

75

iIEIi.

IN=I

4EIX

N

CA

LL

S

MT

l1L

N(X

N,Y

N,N

NN

N,C

E,L

SIG

) IF

(NC

OU

.GT

.1)G

O

TO

3

0

IF(N

tiX

II.:

IE.

1)

GO

T

O

90

13

X

RIG

(l)=

XtI

(l)

XR

IG(Z

)=X

N(l

) Y

RIG

(l)=

YtI

(I)

YR

IG(2

)-Y

tI(1

) G

O

TO

3

11

1

DO

3

1 I

=l,

tIN

XN

X

RIG

(I)=

XN

(I)

YR

IG(I

)=Y

N(I

) LR

=I.IN

XII

ED

lX=

XN

(1

) E

DlY

=Y

N(l

) E

DZX

=XN

(!II

:XN

) E

DZ

Y=Y

N(N

NX

N)

GO

T

O

35

C

AL

L

RI D

GE(

XN,Y

N,Z

N,

NN

XN

)

Figure 1.13

COWER Continued

IF(NCCU.liE.2)

GO T

O 372

CALL

SOilTER(XRIG,NX,LR)

DO 3

71 I=

l,LR

EOlY

=YRI

G(l)

GO

TO 3497

I PEN.3

CALL PL

OT(E

DlX,

EDl

Y, I PE

N)

I PEII=2

CALL

PLOT(XRIG(l),YRIG(l),

IPEN

) ED

li-X

RlGt

l)

EDlY

=YRI

G(l)

GO

TO 3497

- \' c,.3

1=

. FA

LSE.

CA

LL ;I

ORKZ

(XRI

G,YR

IG,L

R,LS

lG)

IF(E

DZX.

EQ.X

RICi

(LR)

.AND

.ED2

Y.EQ

.YRI

G(LR

))

GO T

O 3

498

IF(EIID?)

GO T

O 3

495

EI;DZ=.TRUE.

ED?X

=XRI

G(LR

) EC

ZY=Y

RIG(

LR)

GO T

O 35

I PE:i=2

CALL

PL

OT(E

DZX,

EDZY

, IP

EN)

ED?X

=XRI

G(LR

) ED

ZY=Y

RIG(

LR)

GO TO 3

5 E:iD2=.

F4LS

E.

WK?T

=FKP

T+CE

LLS

IF(LIKPT.LE.XMAX)

GO T

O 2

RETURN

EKD

Fig

ure

1.1

4

CO

WE

R Continued

Ttt

E

PU

RP

OS

E

OF

TH

lS S

UB

RO

UT

INE

IS

TO

F

l tlf)

T

HE

E

DG

E

OF

TH

E

DA

TA

S

ET

IlH

lCH

IS

CLO

SE

ST

TO

A

GIV

EN

L

IiIE

. T

IiIS

S

UB

RO

LJTI

tIE

I

S U

SE

D

WH

EN

A

Llt

lE

IS P

AS

SE

D TC

S

Ce'

iET

:I

~+

ICH

Pr,

ssE

s D

IRE

CT

LY

TH

RC

UG

II A

DA

TA

P

OIN

T.

OB

VIO

US

LY

IF I

T I

S O

HLY

Kt

:O!.l

i T

HA

T

TH

E

LIi

IE

IS S

TR

AIG

tIT

Tr

,EtJ

I

T

IS I

f4P

OS

SIB

LE

TO

K

hiO

Il W

HIC

H

ED

GE

S

OF

Ttl

E

TR

IAII

GU

LAR

I:E

TWO

RK

TH

E

LI'

IE

EX

ITS

Tl

ll'IO

ULH

, S

IHC

E

AL

L

TRI,'

i!IG

LES

OF

T

HlS

GR

OU

P

HA

VE

T

HlS

OR

E

D,IT

A

PO

I?iT

AS

A

V

EK

TIC

E

OF

EA

CH

. TH

US

, I!r

l&T

TH

lS

SU

ER

0UT

I;IE

D

OE

S I

S T

O

Flk

D T

HE

A

NG

ULA

RLY

A

DJA

CE

hT

ED

GE

TO

T

HE

L

I:iE

Ill

QU

ES

TIO

II.

Xl,Y

l,XZ

,Y2

T11E

X

-Y

C00

RD

ll.iA

TE

S

OF

TH

E

EN

DP

OIN

TS

O

F

TH

E

LIN

E

I r: Q

UE

ST

IOII

. x,

y A

RE

V

EC

TOR

S

OF

LEiI

GT

II

N

CO

t!T

AIt

!IN

G

TH

E

X-Y

C

00

R0

1tl

AT

ES

OF

T

HE

D

AT

A

SE

T.

ti P

T

A 7

3Y

H

AI!

R\Y

O

F T

HE

P

Ol!.

TE

RS

O

F T

HE

D

AT

A S

ET

. tI

T

HE

II

U:'S

Ef?

O

f

DO

INT

S

1:' T

taE

DA

TA

S

ET

. 'I

R

Tl!F

L I:I

!:lC

ER

OF

Tli

E

DA

TA

P

OIX

T I

:HIC

II

.IS

B

EIN

G

PA

SS

ED

T

IIR

OU

G'~

. P

~A

N

TH

E

PO

SIT

IO:I

O

F

TH

E

ADJA

CE:

IT

PO

INT

ER

IN

N

R'S

L

IST

T

Hl S

I S

Il

liA

T A

tIG

FN

D

RE

TUR

GS

.

SU

BR

OU

TI I

IE A

NC

FND

(X1,

X2,

Y1,

Y2,

X,Y

,NP

T,N

,NR

,NA

ti)

DA

TA

?

1/3

.1~

?5

93

/ -

CO

':':O

i!/A!

:GLE

/ LL

AB

, LA

B,T

ES

T,t

XL,

CA

L LO

G1

CA

L LL

AB

,TE

ST

,CA

L,A

NG

LL,A

WE

--

-

r,:=Y

?-Y:

T

AX

G-A

TA

N2

(Ala

A2

)*

(-1. )+Pi02

IF (T

AN

G.

LT.

0 T

AN

G-P

I 2*

TA

NG

J=

l co

TO

5

IF(l

It~

'i.i

Q.O

)GiR

lTE

(6,1

00

) F

OR

II,?

T('

','

AN

GFE

;D

DID

NO

T W

OR

K')

tI

R-K

IF

(P:R

.?!E

.L)

RE

TU

RN

L;

lL =

LA:,C

LL

:\b=

+lN

GLL

T

CS

T-A

NG

TE

R

ET

UR

N

Eli

D

TH

E

PU

2PO

SE

O

F T

HlS

S

UB

R5U

TIN

E

IS T

O

INS

ER

T

PO

INT

S A

T A

G

IVE

N

DIS

TA

NC

E

FRO

M

TH

E

PO

INT

S A

LON

G

A

CO

NTO

UR

L

INE

SO

T

HA

T

TH

E

PR

OG

RA

If S

MT

HLN

H

ILL

MO

T C

AU

SE

C

ON

TOU

RS

TO

C

RO

SS

, F32

A

MO

RE

CO

iIPLE

TE

D

ES

CR

IPT

ION

O

F T

HIS

SE

E

TH

E

P2O

GR

AM

W

RIT

E-U

P.

TH

E

SE

CO

tlDA

RY

P

UR

PO

SE

O

F T

HlS

S

US

RO

UT

INE

IS

TO

C

ALC

ULA

TE

T

HE

P

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APPENDIX T I : A SAMPLE DATA STRUCTURE

A sample d a t a s t r u c t u r e was prepared f o r t h e topographic s u r f a c e of

t h e Ptarmigan Creek a r e a of Canoe River v a l l e y l oca t ed i n c e n t r a l

B r i t i s h Columbia (F igure 4 . 1 ) . This i s a good s u r f a c e f o r tests of

t h i s d a t a s t r u c t u r e s i n c e i t c o n s i s t s of r e g u l a r s u r f a c e f u n c t i o n s

i n t e r s p e r s e d wi th i r r e g u l a r func t ions . The r i v e r bottom i s a w e l l

def ined g e n t l y s lop ing p l a n e w i t h t h e s i d e s of t h e v a l l e y breaking

s h a r p l y from t h e bottom and r i s i n g t o i r r e g u l a r peaks. Surface

s p e c i f i c p o i n t s were chosen from t h e s u r f a c e a s p o i n t s r e p r e s e n t a t i v e

of changes i n neighbourhoods. A t r i a n g u l a t i o n of t h e s e p o i n t s was

c a r r i e d o u t wi th t h e c r i t e r i a f o r t r i a n g l e formation being t o minimize

t h e d i s t a n c e between t h e t r i a n g u l a r p l ane and t h e r e a l s u r f a c e

(F igure 3.3). The v e r t i c e s of t h e t r i a n g l e s were numbered from one

through e i g h t y one a s shown i n F igure 11.1. Following t h e r u l e s

o u t l i n e d i n CHAPTER 111 t he d a t a were recorded i n t h e format

The f i n i s h e d d a t a s t r u c t u r e , ready f o r u s e w i t h t h e d i s p l a y r o u t i n e s ,

is shown i n Figure 1 1 . 2 . This s t r u c t u r e c o n s i s t s of e i g h t y one r e a l

d a t a p o i n t s and e i g h t dummy d a t a p o i n t s ( i . e . e i g h t r e a l p o i n t s have

more than seven p o i n t e r s ) . This d a t a s t r u c t u r e was used t o produce

a l l t h e example g raph ic s of t h e d i s p l a y system o u t l i n e d i n t h i s t h e s i s .

C8

Figure 11.1 Irregular Data Structure. The triangulated surface specific points ready for recording the data.

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Figure 11.2 Records of the Data Structure

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