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This is to certify that the
thesis entitled

MEASURING AND MODELLING PERCEPTUAL SHAPE DISTORTION
ON MAP PROJECTIONS

presented by

Dawn Elizabeth Carlson

has been accepted towards fulfillment
of the requirements for

MASTERS degree in GEOGRAPHY

 

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MEASURING AND MODELLING PERCEPTUAL SHAPE DISTORTION
ON MAP PROJECTIONS

By

Dawn Elizabeth Carlson

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF ARTS

Department of Geography

1992

 

 

 

 

ABSTRACT

MEASURING AND MODELLING PERCEPTUAL SHAPE DISTORTION
ON MAP PROJECTIONS

by

Dawn Elizabeth Carlson

Shape distortion is a concern for cartographers because they wish to
communicate spatial information as accurately as possible to map viewers and shape
is a fundamental visual property of spatial entities on a map. Currently, no
techniques or standards exist for measuring shape distortion or selecting map
projections with minimal amounts of shape distortion. This study analyzed three
simple shape distortion indices for their ability to approximate amount of perceived

shape distortion. Thirty three subjects evaluated the amount of visible distortion on

 

sixteen outlines of each of three continents. Their scores were compared to the

index values. The index that best approximated them was a radial deviation index

which was then used to produce a model of perceptual shape distortion to find

critical values for acceptable shapes on maps.

ACKNOWLEDGEMENTS

I would like to thank my advisor, Dr. Judy Olson, for all of her encouragement and
guidance, and for being flexible to meet at odd hours to work around my schedule. I thank
Dr. Richard Groop for his helpful suggestions and criticisms. I am also indebted to the
Department of Geography for providing financial support and valuable experience
throughout my graduate endeavor.

Several others deserve recognition. Mike Lipsey and Ellen White freely shared
their time and expertise. Bruce Pigozzi reldndled my math interest. Sharon Ruggles and
Marilyn Bria were always willing to answer my question, and my fellow classmates

shared with me many hours of discussion, exploration, and discovery. I thank you all.

iii

 

 

 

 

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................... v
LIST OF FIGURES ............................................................................................................ vi
CHAPTER:
I. INTRODUCTION AND STATEMENT OF THE PROBLEM ................ 1
Statement of Problem ....................................................................... 7
H. TECHNIQUES FOR SHAPE ANALYSIS ............................................... 8
Shape indices .................................................................................... 8
Shape change measures .................................................................. 14
Minimal shape distortion projections .............................................. 16
III. METHODS ................................................................................................ 20
Outline production .......................................................................... 20
Shape distortion indices .................................................................. 25
Perceptual evaluation ...................................................................... 27
IV. ANALYSIS AND RESULTS ................................................................... 31
Perceptual results ............................................................................ 32
Index values as predictors of perceptual scores .............................. 41
Summary ........................................................................................ 48

V. DISCUSSION AND SUGGESTIONS FOR FURTHER RESEARCH.49

Suggestions for further research ..................................................... 52
VI. SUMMARY .............................................................................................. 53
Appendix:
A. Programs for Calculating Shape Distortion Index Values .......................... 57
B. Materials for Subject Testing ...................................................................... 68
REFERENCES ................................................................................................................. 85
iv

 

 

TABLE:

S”

#9.“?

 

LIST OF TABLES

List of projections used to produce distorted landmass outlines ................. 21
List of origins used to produce distorted continental outlines ..................... 23

Frequency of Correlation Coefficients for perceptual scores between

subjects ........................................................................................... 35
Perceptual distortion categories for each landmass ..................................... 38
Perceptual distortion categories for combined landmasses ......................... 40
Correlation coefficients between perceptual scores and index values .......... 43

Linear regression results for an equation to model perceptual shape
distortion from index values ........................................................... 46
Predicted critical index values for acceptable and unacceptable shape

distortion ......................................................................................... 47

 

LIST OF FIGURES

FIGURES:
1. Snyder’s minimum-error pseudocylindrical equal-area projection with pole
line .................................................................................................... 4
2. Aribert Peter’s distance-related world map ................................................... 5
3. Bunge’s vertex-lag method for shape analysis ........................................... 11
4. Medial axes approach to shape analysis ...................................................... 13
5. Biorthogonal grid analysis of shape transformation ................................... l7
6. Graphs of perceptual score distributions for each distorted outline grouped
by landmass .................................................................................... 33
7. Graphs combining all landmass perceptual score distributions .................. 34
8. Plots of median perceptual scores against radial deviation index values ..... 44
vi

 

CHAPTER I

INTRODUCTION AND

STATEMENT OF THE PROBLEM

Italy is boot-shaped; Japan is a curved chain of islands; Mexico is shaped like
a leg of lamb! Shapes are important for recognizing places on maps. The National
Geographic Society recognized the importance of shape when it selected the
Robinson projection for its official world map. Among the reasons the Society chose
this projection was that: "In the combination of shape and area it matches reality
more closely than its venerable predecessors" (Garver 1988).

Even though cartographers and the National Geographic Society consider
shapes important, continental and country shapes are seldom discussed in the map
projection selection literature. Current guides discuss the preserved properties of the
projection (equal area, conformal, rectilinear great circles, etc...) and the patterns of
parallels and meridians resulting from the geometric form on which it is developed
(conic, cylindrical, etc...), but the distortion discussed in these guides is often the
distortion of the latitude/longitude grid, not the distortion of continental shapes.

Even new selection guides do not discuss these shapes explicitly. For example,
the American Cartographic Association recently published a booklet entitled
"Matching the Map Projection to the Need" (American Congress on Surveying and

Mapping, 1991). Many map purposes and projections for those purposes are

2
discussed, but none of the authors mention continental or country shapes. Neither
do Nyerges and Jankowski (1989) in their article entitled "A Knowledge Base for
Map Projection Selection," which attempts to set up a "smart" computer system to
allow computers to select an appropriate map projection for a map.

If such a system ignores shape, how can it select the best projection? A map
must look right to be trusted and currently, the only way to judge the amount of
shape distortion on a map is to visually compare shapes to those on a globe. As
computer software for plotting projections makes more projections available,
cartographers must be careful in their selection. We cannot just use the first
projection that meets the general distortion requirements. Many different projections
are equal area or conformal or cylindrical or conic but some projections represent
shapes better than others.

Since there are no guidelines for allowable amounts of shape distortion in the
selection process, the evaluation of this distortion is left to the subjectivity of the
cartographer. Each cartographer must examine the shapes on the projection and
accept or reject those shapes. This evaluation is difficult because some areas may
be extremely distorted while other areas are not and some areas may be distorted but
still communicate shape information effectively. A quantitative evaluation technique
coupled with perceptive verification is needed to assist cartographers in the shape
evaluation process.

The question faced is: how can shape distortion be measured? Tissot’s well-

known theory of distortion only measures angular and area] distortions at points on

3

the map. Authors often state that maps with zero angular distortion (conformal
maps) preserve shape (Espenshade, 1986: p.x; Robinson, 1984: p.83; Muehrcke,
1978), but this is not true as Dent (1987) clearly states. He uses the Mercator
projection as an example.

"Nothing could be further from the truth; in fact, land masses

do not stretch to infinity as they appear to do on the Mercator

projection, and even more, the poles cannot be represented. For most

geographic mapping purposes selecting a world conformal projection

so that continental shapes are preserved is meaningless."

Equal area maps, maps with no areal distortion, are often used at small scales
because they show correct relative size. But since the geometrical nature of
equivalency requires that scales vary in different directions about a point, shapes are
greatly distorted on these maps. When shape is important to map communication,
both angular and areal distortion must be sacrificed (Canters, 1989; Dent, 1987).

Minimum error projections balance the angular and areal distortions but still
represent shape imperfectly. Snyder’s minimum-error projection is one such example
(Figure 1). Some other minimum error maps minimize linear distortion measures
rather than Tissot’s measures (Snyder, 1985; Peters, 1984). Linear distortion is the
deviation between distances on a map and their corresponding globe distances. The
mean of these values is used to summarize the amount of linear distortion on maps.
But even though Peters (1984) claims that, "The true sizes and shapes of most
continents are more closely approached on a distance-related map than on other

maps," the shapes of continents on his map projection look as distorted as they do

on many other projections (Figure 2).

 

           

       

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Figure 2. Ariben Peters’ distance-related world map.

 

6

Shape is a perceptual phenomenon and is difficult to define and measure in
a mathematical or analytical sense. Scientists in a variety of fields have developed
and continue to develop definitions of shape and methods for classifying shapes
based on indices, but the indices are not based on perceptual judgments of shape.

Shape indices represent shape with a single value that is derived from a
specific combination of measurements of a figure. An index is considered "good" if
similar shapes have similar values. Unfortunately, these indices are limited to
measuring one or two aspects of shape such as elongation, compactness, surface
indentation, or dissection. No one index can evaluate all of these shape aspects.

Another drawback in using shape indices is that they do not measure
distortion, or shape change. If a shape is compared to a distortion of the same
shape, the indices would indicate that they are entirely different shapes instead of
two representations of the same shape. The human eye can see the relationship, but
the mathematical indices cannot.

Biologists have made some progress in analyzing shape change, but their
methods were developed to study differential changes in shape. Biologists want to
pinpoint where one shape differs from another or how one shape "grew" into another.
Interested in tracking the grth of an organism, they ask: Which bones grow first?
Does the organism increase in width as it increases in length? Do the fins on a fish
change position as the fish ages? (Bookstein, 1985)

Cartographers, on the other hand, are more concerned with overall amounts

of shape distortion or shape change or the magnitude of difference between a map

 

7

shape and a globe shape. We want to know how much change can occur before the
shape becomes unrecognizable, or how difficult it is to recognize shape when it is
distorted. We are particularly concerned, in other words, about any change that
significantly affects map communication.

For cartographers to study shape change, a shape change index needs to be
developed and it must measure overall amounts of shape change, not pinpoint exactly
where the change occurs. It must also reflect perceptual shape change in its
measurement; to quote Canters (1989),"... any attempt to develop adequate world
maps by quantitative techniques can only be successful if perceptive requirements

are taken into account during the design process."

Statement of Problem

The problem facing cartographers is that there is no commonly-used method
for analyzing shape distortion on maps and there are no guidelines for objectively
selecting maps with allowable amounts of shape distortion. The decision is left solely
to the experience and preference of the cartographer.

The following research attempts to modify three existing shape indices into
shape change or shape distortion indices and then answer the following two
questions: Do these shape distortion indices predict the amount of overall shape
distortion perceived? What are the limits of allowable distortion in shape, i.e., how
much shape distortion as measured by the indices does it take to render a shape a

"bad representation"?

 

CHAPTER II

TECHNIQUES FOR SHAPE ANALYSIS

Shape analysis has been used and studied in a wide range of disciplines from
topology in which shape theory is the concern, to biology in which growth of
organisms is studied, to geology in which the shapes of sand grains are analyzed to
remote sensing where shape recognition is the focus. Every discipline concentrates
on different dimensions of shape such as compactness, elongation, dissection, surface
indentation, or differential change and different shape indices have been developed
for each of these disciplines.

In this chapter, I will limit my discussion to shape indices developed for
measuring two-dimensional figures since shapes on maps are two-dimensional. The
indices increase in complexity beginning with those for measuring and analyzing
shape, which are primarily used in shape classification, and ending with indices of
shape change, which are extensions of those shape indices used for classification. At
the end of the chapter I will discuss previous attempts to produce world maps that

reduce the amount of shape distortion.

mlpLiLdigcn

The simplest shape index is the ratio of perimeter to area. It was first

proposed in 1822 and is still used today. Generally, the formula for the index is I =

 

9

k ' P/A where I is the index, k is an arbitrary constant, and P and A are perimeter
and area respectively. Unfortunately, its simplicity does not outweigh its problems.
The length of the perimeter relative to area is only a measure of surface indentation.
Two figures having almost identical shapes but differing perimeters (a smooth boot
shape versus the same boot shape with numerous minor indentation and protrusions)
will have widely different index values, implying that they should be considered as
very different shapes (Frolov, 1975).

Another type of index includes those derived from the linear dimensions of
a figure and its area. These include: 1) the ratio of width and length to show
elongation, 2) the ratio of the square of the length of the long axis to the area, and
3) the ratio of the length of the long axis to the diameter of a circle having the same
area as the figure. Again, these indices are very simple and only describe limited
characteristics of the shape. Frolov (1975) summarizes this group by saying that

"various combinations of length, width, and area of a figure of arbitrary

configuration do not yield new information about shape, do not

represent independent parameters of shape, (like compactness,

indentation, and dissection) and are therefore of limited usefulness."

A third type of shape index is derived from the dimensions of inscribing and
circumscribing circles. The diameter of the largest circle that can be inscribed and
the diameter of the smallest circumscribing circle are used as the primary

measurements of shape. But, these measurements are the same as measuring the

longest and shortest axis which brings us back to the problems mentioned in the

previous paragraph.

 

 

 

10

Tangents to a figure were also used to describe a shape according to its
degree of dissection. A tangent "sliding" around the outside of a figure is an
interesting idea, but what is measured and recorded was not explained well and the
technique was never further developed (Frolov, 1975). Bookstein (1989) discusses
the use of tangent angle which may be a similar measure but his discussion is
confusing as well. An index that measures radial distances from a center of a
figure to the circumference at equal intervals of angles was proposed by Boyce and
Clark in 1964. Tire measure was developed to analyze shape compactness and
surface indentation but it has one prominent flaw. Equally spaced radials do not
intersect the circumference at equal intervals so some of the circumference is
weighted more than the rest. Thus, the index represents some parts of the figure -
the smoothest part - and not the entire figure. Again, the same index value may be
attributed to two very different shapes.

Bunge (in Boyce & Clark, 1964) developed an entirely different approach to
analyzing shape (Figure 3). He assigned a set of six numbers to inscribed octagons
with equal sides. The six numbers identified the shape. To obtain the numbers, one
vertex in the polygon was assigned the role of origin, the place where measurements
were to begin. The first of the six numbers was the sum of the distances between
every other of the octagon’s vertices. One vertex was skipped with every
measurement. The second number measures distances between vertices skipping

two. The third skips three, the forth, four, and so on.

 

 

 

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12

Unfortunately, Bunge’s imaginative technique suffers from shortcomings as
well. First of all, it describes the polygons, not the figure. Also, measurements
starting with different vertices give different values and a figure may be approximated
by different polygons. Most importantly, he never explained how to use all six
measurements in shape analysis.

Pavlidis (1978) suggested some other algorithms for shape analysis. Among
them are medial axes and shape decomposition for shape recognition. In the medial
axes technique, a figure is transformed into a line drawing or skeleton of the original
figure (Figure 4). The medial axes, as Pavlidis defined them, are found in the

following manner:

"Let S be a set in the plane and let B be its boundary. If X is a point

which belongs to S then it is always possible to find its closest neighbor

belonging to B. If X has more than one such neighbor then it is said

to belong to the medial axis of S."

The "skeletons" of figures can be compared by measuring lengths of and angles
between axes by superimposing one on another (Bookstein, 1985). If the "skeletons"
are similar, the figures are considered to have essentially the same shape. This
method can be very time consuming, especially for complicated structures such as the
shapes of continents. It is also primarily concerned with testing the similarity of
shapes as are all methods of shape recognition. Shape distortion, or shape change,

on the other hand, is more concerned with finding the differences between similar

shapes.

 

 

 

13

 

Figure 4.. Media] axes approach to shape analysis.

 

 

14
Shag change measures

Analyzing shape change requires a different set of indices than the ones used
for shape classification or pattern recognition mentioned above. Shape change
investigates figures that have the same general shape but appear to be tilted or
skewed or stretched due to deviations between vertices and other landmarks.

Biologists, in using shape change measures to study the growth of organisms
and the possible evolution of one organism into another, have interests that are
similar to cartographers concerned with shape distortion. Geographers and
cartographers are interested in the deviations of a sample shape from a original or
"true" shape, or the deviation of shapes on one map from the shapes on another.
Because the indices used by both groups are similar, I will use examples from both
to explain the various types of shape change indices.

The most basic method of analyzing shape change is to overlay or trace one
shape onto another. The differences between them can be readily seen and
deviations of landmarks can be easily measured. However, Bookstein (1989)
emphasized the fact that measurement error is bound to occur as well as registration
problems. If no pre-existing knowledge of growth and growth rates exist, registration
cannot be done correctly and the results are highly questionable. Also, conclusions
vary with the choice of registration.

Other problems that can occur in overlay analysis are differences in scale and
orientation. In biology, organism may vary greatly in size and still have similar

shapes. In cartography, maps can vary in scale and orientation, making overlays

 

 

15

particularly difficult and arbitrary. To alleviate these problems, Waterman and
Gordon (1984) used a least-squares Euclidean transformation. This procedure
neutralized the effect of rotation, translation and scale change in their analysis of
mental maps. The coordinates of landmarks in the mental maps that were drawn by
subjects were transformed so that the sum of squares of the distances from the "true"
map to the "transformed" map was minimized. The transformation gave the best "fit"
possible. However, the index only measures the distortion among the positions of the
landmarks and not of the entire map.

Tobler (1986) discusses a method for measuring the similarities between map
projections. In a sense, he used a type of overlay analysis. The similarity of two map
projections can be found by calculating the lengths of the vectors connecting
corresponding points. The root mean square distance of all the lengths is the index
of similarity (or dissimilarity). He also extended this method to include the
measurement of the departure of a map from the sphere by measuring "strings" going
from the globe to the flat map. The globe was positioned optimally on the map
using a least squares fit to minimize the length of the "strings". Unfortunately the
mathematics get very complicated. It is also not clear that this technique gives any
information about the deviations of shapes on the map from those on the globe, but
instead demonstrated how much the projection deviates from the sphere.

Sharp (1971) does suggest a method for measuring shape distortion that uses
a ratio of Tissot’s scale factors as an index of shape distortion at a point on the

projection. Unfortunately, his index brings us back to the problem of analyzing

 

 

16

infinitely small areas on a map when we are really concerned with distortion over
large, finite areas that affect perception.

Biorthogonal grids are another method for analyzing shape distortion (Figure
5). This technique reduces changes of shape to differential changes in size. It tracks
the stretches and shrinks of the internal segments of a grid imposed on the figure.
In the end, information about growth is extracted by matching little cells in a grid to
its transformed image (Bookstein 1989).

What is fascinating to a cartographer about biorthogonal grids is that they are
based on the same theorems as Tissot’s indicatrix. In fact, biorthogonal grids are
produced in the same manner as the indicatrix, and therefore have the same problem
for cartographers. They only measure rates of change at a point and are not a
method for summarizing overall distortion.

Cartographers need one value to summarize shape distortion. With such a
value, projections that minimize continental (or country) shape distortion on world
maps can be produced. Several cartographers have attempted to produce maps with

low shape distortion but with varying methods and results.

Minimal Shape Distortion Projection

Probably the clearest attempt is the projection by Aribert Peters developed
in 1984. He minimized random distance distortions about the globe, reducing the
amount of angular distortion on the map thereby deducing that shapes on this

projection are better represented than on any other non-interrupted projection.

 

 

17

 
 
 

 
 
  
 
  
  

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et. al. 1985. WWW. Special Publication 15,The

Academy of Sciences of Philadelphia. pp. 129,139.)

 

 

 

18

Dent (1987) used an entirely different approach. He developed a composite
world map of several orthographic projections, one for each continent. Subjects
chose the "center" of a continent and these "visual centers" were used as the origin
for each orthographic view. The views were pieced together to produce a world map.
This "Poly-Centered Oblique Orthographic World Projection" represents the
continents very well, but the discontinuous appearance of the map is distracting.

Maps that represent the earth continuously are much more appealing.
Robinson’s projection as well as projections developed by Baranyi (1968) and Canters
(1989) show the world on whole, non-disjointed maps. Robinson and Baranyi used
an iterative plotting process that was repeated until the shapes of the landmasses
satisfactorily approximated the true globe shapes. Canters added a quantitative
technique to the iterations to systematically refine the projections by adding more
and more constraints to the appearance of the maps. He is convincing in his
criticisms of iterative techniques that do not use a quantitative evaluation in the
development of a projection by arguing that

"the final map is to a very large extent the result of the designer’s

experience. It has no mathematical formulas in the usual sense...it may

be argued whether the total reliability on the cartographer’s perception

does not introduce too much subjectivity in the designing

process...Therefore it is the author’s opinion that most advantage can

be derived from the coupling of a quantitative evaluation technique

with a process of perceptive verification. In this way subjectivity is

limited to the introduction of a number of appropriate constraints

defining the general appearance of the graticule. It must however be

admitted that a careful definition of these constraints remains a
necessary condition for the development of acceptable maps."

 

 

19

Canters’ mean linear distortion measure is similar to Peters except that shorter
distances for the analysis were used because the mean linear distortion value of a
projection substantially decreases as distance increases. The projection equations can
be constrained to allow certain map characteristics to be maintained such as
symmetry about the central meridian, equally spaced parallels and the representation
of the pole as a line. More constraints can be imposed to produce different looking
maps which all have minimal amounts of distortion.

The production of projections that minimize shape distortion needs standards,
but as yet no method exists for adequately measuring shape distortion. If such a
method could be found, and if that method incorporates the perception of shape
distortion, new standards for projections that minimize shape distortion could be
developed and the resulting maps would best represent the continents relative to
their true shapes.

I selected three shape indices - a perimeter to area index, a grid segment
deviation index, and a radial deviation index - to modify and then test for their ability
to approximate peoples’ perception of amount of distortion. The indices were chosen
from the shape literature as a selection of highly varying and yet simple methods and
were modified so that they could be used on a spherical globe surface as well as on

a two-dimensional map. The next chapter describes the methods used to evaluate

them.

 

 

CHAPTER III

4 METHODS

To study the relationship between perceptual shape distortion and shape
distortion measured by mathematical indices, many outline maps of Australia,
Greenland, and Africa were produced using WORLD, a projection and plotting
program (Voxland, 1987). These three landmasses are located at different latitudes
and vary in size so that a wide range of distortions could be analyzed. Tire outline
maps for each landmass varied in their projection and origin, and in the amount of
shape distortion measured by the three shape indices: ratio of perimeter to area,
mean linear deviation, and mean radial deviation. University students visually
evaluated and scored the landmass outlines for shape distortion. The relationship
between their perceptual scores and the index values could then be studied using

median graphs, correlation and regression. Each step is discussed in detail below.

Outline production

Twenty projections that are most often discussed in the projection selection
literature were used to produce the distorted landmasses (Table 1). The selection
includes all of the major forms of projections available (cylindrical, conic, azimuthal,
polyconic, pseudocylindric, pseudoconic) and all three major properties (equal area,

conformal, neither). Origins for these projections were selected from twenty regions

20

 

21
Table l

Projections used to produce distorted landmass outlines

Mercator

Transverse Mercator

Platte Carre
Miller

Albers
Kavraisky No. 4
Lambert

Equidistant
Gnomonic

Orthographic
Stereographic

Hammer-Aitoff
"American"

Van Der Grinten
Winkel Tripel

Goode's
Mollweide
Sinusoidal
Robinson

Bonne

Cylindric
Cylindric
Cylindric
Cylindric

Conic
Conic
Conrc

Azirnuthal
Azimuthal
Azimuthal
Azimuthal

Polyconic
Polyconic
Polyconic
Polyconic

Pseudocylindric
Pseudocylindric
Pseudocylindric
Pseudocylindric

Pseudoconic

Ellipsoidal Conformal
Ellipsoidal Conformal
Equidistant

Equal Area
Equidistant
Conformal

Equidistant from center
Great Circles rectilinear

Conformal

Equal Area

Ellipsoidal

Global circular grid

Eq. spaced on central merid.

Homolosine Equal Area
Elliptic Equal Area
Equal Area

 

Equal Area

 

 

22

on the globe in a stratified random sampling system. The globe was divided into five
segments of longitude (72 degree spans) starting with zero, and into four segments
of latitude (45 degree spans). Using a random number table, origins were selected
within each region so formed.

Each projection was randomly assigned one of the twenty selected origins by
drawing origins from a hat. Once removed, an origin was not put back until drawing
was complete for that landmass. Because some projections are limited to mapping
only a certain arc distance around the origin, an assigned origin occasionally was
incompatible with the landmass to be mapped. For example, an azimuthal map
centered at 70N, 50W would not show the continent of Australia as a closed unit.
When such an event occurred, another origin was randomly assigned until a usable
one was found. In some cases a new origin had to be selected for one of the regions.

Table 2 lists the regions with their corresponding latitude and longitude ranges
and the origin that was selected for each region. It also lists the projection (indicated
by the first four characters of the projection names in Table 1) that was used with
each origin to produce the distorted landmass outlines. When a selected origin was
incompatible with a projection, I first tried to find another origin from the selected
list (Selected Origins). If none of those worked, I selected an alternate within the
same region using the random number table once again. These are shown in Table
2 in the Alternate Origins Columns. The projection numbers with asterisks (‘) in
Table 2 were dropped later in the study because a pilot test showed that twenty

outlines for each landmass was too many for subjects for analyze.

 

 

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Greenland was the most diffith because of its size and location on the globe.
Some origin regions would not work with the projections resulting in two projections
assigned to the same region in three cases. To keep the outlines for Greenland from
becoming too small and the outlines for Africa from becoming too large, the scales
for the landmasses were different. All of Australia’s outlines were produced at
1:63,000,000, all of Africa’s outlines at 1:100,000,000, and all of Greenland’s outlines
at 1:40,000,000. With these scales, the outlines could be printed on 1/4 of a sheet
of paper, and the set could be handled within the space available to individual
participants.

Cardstock was used as a base for the outlines because its extra strength would
keep the outlines from getting bent and crumpled by the subjects. The cardstock also
allowed the sets of maps to be shuffled before each student’s evaluation.

Coordinate data used for the outlines were from World Data Bank 11, which
is included with the WORLD projection program. They are in POLYVRT chain
data files: CONT96 for Australia, CONT94 for Africa, and CONT91 for Greenland.
Because I needed to compare shapes on the globe to shapes on the map, latitude and
longitude coordinates were obtained by saving WORLD "outform" files after
displaying the lines of interest on the screen. The postscript files produced by
WORLD were also saved and were used to print the distorted outlines as well as to

calculate the shape distortion indices.

 

 

Shape distortion indices

The first of my three shape distortion indices was simply a ratio of perimeter
to area. The difference between the outline ratio and the globe ratio was the
indicator of shape change. Areas for the landmasses on the earth were taken from
the New York Times Atlas of the World, 1991. This index is conceptually different
from the other two. It is easy to measure and if it were effective could easily be put
to use. For the second index, a rectangular grid of points, the intersections of
latitude and longitude lines at five degree intervals, was overlain on the landmass to
be mapped. The five degree interval just outside of the landmass’s border was used
as the perimeter of the rectangle so that the rectangular grid covered the entire
landmass.

Distances between neighboring grid points were calculated on the globe (arc
distances) and on the map (Euclidean distances) and the mean deviation between the
map distances and the corresponding globe distances were recorded. The grid is a
conceptually different way of analyzing shape than the radial index method. This
grid deviation method was used in 1977 by Tobler to produce a minimum-error
projection (in Snyder, 1985), although he did not make any claims to its ability to
capture shape distortion.

The third index was a radial measure based on Boyce and Clark’s index
(1964). Recall that Boyce and Clark measured the distances from a figure’s center
of gravity to the perimeter of the figure along equally-spaced radii and turned those

distances into percentages of the sum of the radial distances. I changed this measure

 

 

26

by using the distances along radii from equidistant points on the perimeter to the
geographic center of those equidistant points. The change from equally spaced radii
to equidistant points around the perimeter was suggested by Frolov (1975) to
improve the index. Equally-spaced points capture more of the perimeter and
therefore the index contains more shape information. The equidistance was based
on the digitized points and only approximates equal distances around the perimeters
of these landmasses on the earth’s surface.

The mean radial distance of the map outlines was used to standardize the
outline radii, and the mean radial distance of the globe measurements was used to
standardize the globe radii. In other words, if a given point was 1.5 standard
deviations on the map, it was 1.5 times the mean radial distance on the map; if it was
1.3 standard deviations on the globe, it was 1.3 times the mean radial distance on the
globe. The root mean square error between standard distances on the map and
globe served as the index of shape change.

All three indices were calculated for each shape outline using QuickBasic.
Appendix A contains copies of the programs. The distance formulas used were the
common ones for measuring arc distances on a globe (AD) and linear distances on
a plane (D):

AD = R ‘ ZU

where cosZu = sinAl'sinAJ + cosAl‘cosAj‘cos(Bl - BJ).

and A= latitude in radians

B= longitude in radians

 

 

27
DD = [(XI 'XJ)2 + (Yr ' YJ)2]l/2

where X and Y are map coordinates

Because QuickBASIC does not have an arccosine option, the arc distance
formula was written in terms of arctangent (Beyer, 1984). The result is this formula

where cosZU is the same as above.

AD = R ' [(pi/Z) - arctan( coszU / (1 - cosZU2)l/2)]

Perceptual evaluation

Originally, subjects were to evaluate all twenty outlines for each landmass.
However, a pilot test indicated that twenty was too many. The set was reduced to
sixteen for each landmass by ordering the shapes according to the mean perceptual
scores obtained in the pilot test and eliminating one shape from each quartile.
Appendix B contains all the materials used in the resulting main test.

Each outline was labelled with a three letter code to keep track of the
mapped region, the projection, and the origin used to produce the outline. The
letters were chosen randomly so that they would not influence the subjects’ task of
ordering the maps.

Thirty-three university student volunteers from an upper-level undergraduate
landscape architecture class were asked to evaluate all three sets of sixteen outline

maps. Their experience with maps is concentrated on very large scales, and I had

 

 

28

no reason to believe that they would evaluate landmass shapes andy differently than
the general public (or at least no differently than college students in general).
Subjects were asked to put the outlines in order from least shape distortion on their
left, to most shape distortion on their right. They were supplied with a globe and
could check each outline with the true shape as frequently as they wished. If subjects
felt two shapes appeared to be equally distorted, the shapes could be stacked
together. Size was not to be a factor in their evaluation. They were also told that
there was no correct answer and that they had as much time as needed to complete
the task.

When the subjects completed the ordering task for one landmass, they were
instructed to record their evaluation on an unmarked scale that was 10 inches long.
They were to mark the left end of the scale with the outline code having the least
amount of distortion, the right end with the code of the outline having the most
distortion. Then they were to place a tick on the line for each of the remaining
outlines making sure to label each tick with the code on the map outline. The mark
would indicate the amount of shape distortion in an outline relative to the two
endpoint maps. If two shapes appeared to have similar amounts of distortion, the
marks would be close together. If neighboring shapes varied more greatly in
distortion, the marks would be farther apart.

To facilitate the task, two techniques were suggested. Participants could
assume the table was the scale and arrange the shapes to help them establish the

correct spacing and then mark the scale to resemble what they had on the table. Or

 

 

29

they could find the shape with the medium amount of distortion and put its mark
near the midpoint of the scale. This provided a third reference point to aid in the
placement of the rest of the map outlines and they could continue placing midpoint
maps if they wished.

The last part of the task was to look at each shape starting with the one
having the least amount of distortion and pick out the first shape that the subject felt
was a "bad" representation of the landmass. The question was meant to be subjective
and to leave the interpretation of "bad" to the subject.

When subjects finished evaluating the set for the first landmass, I collected the
maps. The subjects then proceeded with the next set following the same procedures
and then with the third.

Because the order of landmass evaluation might affect results, landmass order
was rotated for successive subjects. If one subject evaluated Australia first, then
Greenland, and finally Africa, the next subject would see Greenland, Africa, and then
Australia. The order of evaluation was recorded for each subject.

Subject evaluation scale markings for each outline were measured to one
twentieth inch precision from the left (least distorted) end of the scale. That end was
designated as zero. These perceptual values were grouped by landmass and
subgrouped by order of evaluation. If a landmass was evaluated first by a subject,
it was given an order value of one; if second, a two; if third, a three.

The total data collected from the testing can be summarized as follows: 33

perceptual scores for each outline grouped by the order of evaluation and by

 

 

30

landmass and 33 perceptual scores of the outlines for each landmass that were
selected as the first "had representation". I had also calculated the radial deviation,

grid deviation, and perimeter-area deviation index values for each shape outline.

 

 

CHAPTER IV

ANALYSIS AND RESULTS

The purpose of my analysis is to find and describe the relationships between
the mathematical shape distortion indices and the subjects’ perceptual scores. Such
a relationship can be used by cartographers to help analyze the shape distortion on
map projections. Index values can be used to predict the amount of perceptual shape
distortion on maps and to help determine if the shapes on maps are acceptable or
unacceptable.

The analysis is divided into two parts. The first part concentrates on the
perceptual results. It begins with a preliminary test to see if the order of landmass
evaluation affected subjects’ responses. Then graphs of the distribution of responses
for each continental outline and Spearman’s correlation are used to determine if
there was consistency among the subjects’ responses. The interquartile ranges
marked on the graphs are used to indicate whether or not outlines were rated
significantly different from one another. Finally, the results for the question
concerning the first "bad representation" of the landmasses are summarized. The
distribution of these scores and their medians are graphed alongside the outline
scores to see where the outlines were most commonly divided into "good" and "bad".
Categories of acceptable and unacceptable amounts of shapes distortion are also

developed based on the median "bad" response. Outlines that were considered

31

 

 

32

"good" 75% of the time and outlines that were considered "bad" 75% of the time are

listed.

The second part of the analysis examines the relationship between perceptual
scores and the measured shape distortion index values. The perceptual scores and
index values are compared visually and analytically to see which index is most closely
related to the perceptual scores. The index that best approximates the perceptual
scores is then used to develop a model for predicting perceptual scores from the
index values. Finally, the critical index values useful in predicting shape acceptability

are calculated using the model.

Perceptual results

The preliminary analysis to see if the order of landmass evaluation affected
subjects’ responses is necessary because the results could affect the rest of the
analysis. ANOVA was used to test the difference of means between order groups.
The results showed no difference in perceptual scores (F=0.430; p=0.655). It can
be assumed that the subjects evaluated each shape independently, i.e., their shape
distortion evaluation did not change as they became more familiar with the task.

To summarize the perceptual results, perceptual score distributions for each
outline were plotted on scales. Figure 6 shows the scales for each landmass stacked
vertically in order of median scores. Tire interquartile range (the two middle

quartiles surrounding the median) are highlighted. Figure 7 shows the graphs

reordered to elirrrinate the landmass grouping.

 

 

 

AUS

GRN

AFR

 

Figure 6. Graphs showing the perceptual score distributions for each outline separated by
landmass. The scores range from 0 to 10 and the highlighted regions represent the two
middle quartiles about the median score.

 

 

 

 

Figure 7. Combined graphs showing the medians and two middle quartiles of subjects
responses for each continental outline.

 

 

 

35

In the figures, a distinct trend is apparent, which indicates that subjects
generally agreed about the amount of distortion in the sample continental shapes.
To further test the consistency among subjects, Spearman’s correlation was run
between subjects resulting in coefficients for every possible paired combination.

Table 3 shows the frequency of the resulting correlation values.

Table 3

Frequency of Correlation Coefficients between
pairs of subject scores.

# of subject pairs per landmass

0.10 - 0.19
0.20 - 0.29
0.30 - 0.39

0.40 - 0.49
0.50 - 0.59
0.60 - 0.69
0.70 - 0.79

0.80 - 0.89
0.90 - 0.99

 

The most common coefficient between subject pairs was 0.8 to 0.89 suggesting
a strong consistency among subjects in the rankings of visible shape distortion in the
continental outlines. Greenland shows the greatest scatter of correlation values. In
other words, some pairs of subjects had highly different rankings for Greenland.

Given the finding of generally strong consistency, we need to examine whether

successively placed maps were distinctly different in their scores. Close examination

 

36

of the graphs indicates that the consistency is not strong enough to significantly
distinguish every shape from every other. The highlighted interquartile ranges in
Figures 6 and 7 overlap between most successive shapes. This demonstrates that the
subjects, as a group, did not distinguish between all of the different amounts of shape
distortion on the outlines. Visual inspection of the graphs suggest that the subjects’
evaluations of these particular samples of maps produced categories of distortion
rather than a distinct sequence.

Also in the figures are graphs labelled BAD. These graphs show the
distributions for the responses to the question of which shape was considered the first
"bad representation" of the landmasses. The highlighted regions of these graphs
represent the interquartile ranges as in the other graphs.

The BAD graphs show where subjects most commonly divided the outlines
into "good" and "bad" shapes, but they do not give a clear idea of which shapes were
considered good or bad most of the time or if subjects considered the same maps as
good and bad. I therefore examined the each subject’s responses individually to see
which outlines were considered good or bad according to their own choice of first
"bad representation". These results are discussed in more detail later, but in general,
subjects agreed on which maps were "good" and which were "bad". Only five outlines
escaped both categories. Also noteworthy is that the same outlines that were
considered "good" and "bad" by 75% of the subjects are the same outlines that are
separated into "good" and "bad" by the median BAD values listed in the graphs

(Figures 6 and 7). These results are consistent with the high correlations between

 

 

37

the pairs of subjects and lend additional support to the development of categories of
perceptual shape distortion. Because most shapes were consistently considered
"good" and "bad" by subjects, it is reasonable that their evaluations of "good" shapes
produced important categories of shape distortion. The median values for the BAD
scores represent critical perceptual values for acceptable amounts of distortion in
shapes. Any shapes with a perceptual score higher than the median BAD values
would be unacceptable and shapes with lower perceptual scores would be acceptable.

Using the median values for the BAD scores and the highlighted regions on
the graphs, I visually divided the shape outlines into perceptual categories of
distortion. The median BAD values were used to divide the perceptual results into
two categories. Any perceptual score greater than the BAD median is considered,
on average, to be a poor representation of the true shape. Perceptual scores less
than the BAD median suggest that the shape is an acceptable or "good"
representation. The other category breaks subdivide those two categories into very
good and good, and bad and very bad.

Even though I developed the categories visually based on interquartile ranges,
the category breaks are also located where distances between the perceptual medians
of successive outlines is great. A least squares or natural breaks classification of the
medians would produce similar categories. The resulting perceptual category breaks
for the landmasses are listed in Table 4 along with the outline codes that fit into
each category for each landmass. The heavy line within the landmass categories

shows where the median BAD value divided the shapes into "good" and "bad". The

 

 

38

Table 4
Perceptual Categories by Landmass

Land- Perceptual Code Median Radial Grid Perimeter
mass Breaks Perceptual Deviation Deviation Area

1 0 It . It . Inx
AUS 0 .-0 0.8 KBG“ 0.00 0.32 0.018 1.04
MIH‘ 0.45 0.37 0.023 1.03
05X" 055 0.01 0.0008 .
URS‘ 1.35 0.18 0.001 0.08

CON‘ 1.40 0.12 0.001 0.22

LYT 2.10 0.31 0.057 1.43
PLV 2.85

2.95 - 75 ‘DUR 4.80 0012 0:47
“CEN 5.30 0. 004 0.13
near 5.70 . 0.005 0.41

0.004 1 .37
- l .67

. I 1
7 .-6 9.5 “YUW 0.33 0.014 0.24
::OMJ 8. 70 1.04 0.015 051

I .,I 1.01
-M_m

. I -_. AAI I '

6.1 - 9.0 ‘:WFQ 6.65 0.74 0.006 .
“ORV 8.00 1.22 0.015 2.47
“DMF 8.35 1.14 0.019 3.24

9.1 - 10.0 “LIX 9.75 0.63 0.207
“SUO 10.00 1.35 0.072

0.005 TIV“ 0.20 0.38 0.467 0.10

01:0" 0 .
TAR" 0.75
UP" 1.30
INK‘ 1.40
DSB' 1.65
‘YTE «

3.5-35 “VON
“YSN
“RMC
ttm
.tSXL
1*.pr
“VIZ

8.6-10.0 “UAM 9.25
"WQP 9.80

** on right indicates that 90% of subjects considered the nrap "'good',
on left indicates that 90% of subjects considered the map' "bad".
* on right indicates that 75% of subjects considered the map "";good
on left indicates that 75% of subjects considered the map "bad".

 

39

codes with asterisks (’) are the continental outlines that were considered "good" or
"bad" 75% and 90% of the time. Two asterisks to the right of the code indicate that
the outline was rated "good" by at least 90% of the subjects; one asterisk indicates
that an outline was rated "good" at least 75% of the time. Asterisks on the left side
of the code indicate which outlines were evaluated as bad by at least 90% (”) and
at least 75%(‘) of the subjects. Also shown in the table are the median perceptual
scores for the shape outlines, and the three calculated index values for each of the
shape outlines. Table 5 is identical except that it shows the categories that I
developed when considering the outlines of all landmasses together.

For Australia the perceptual breaks are 0 to 0.8 for the most acceptable
shapes, 0.9 to 2.9 for acceptable shapes, 2.95 to 7.5 for unacceptable shapes, 7.6 to
9.5 for the most unacceptable shapes, and one outline was considered significantly
unacceptable and was therefore given its own category from 9.5 to 10.0. Greenland’s
breaks were a bit more lenient, i.e., higher median scores for acceptable shapes
allowed more shapes to be considered "very good". The first is 0 to 2.0, then 2.1 t
4.0, 4.1 to 6.0, 6.1 to 9.0, and 9.1 to 10.0. Africa’s outline division resulted in one less
category. The breaks are 0 to 0.5, 0.6 to 3.4, 3.5 to 8.5, and 8.6 to 10.0. For the
combined results, there were four categories. The breaks are 0 to 0.6, 0.7 to 3.3, 3.34
to 7.6, 7.7 to 10.0. The only outline whose 75% ranking puts it into a different
category from its median score is YTE of Africa. The median based categories
consider it "good", but its 75% ranking considers it "bad".

Since the median BAD value is a reasonable division between the 75%

 

 

40

Table 5
Perceptual Categories for Combined Landmasses

1.04
0.10
0.13
0.28
1 .73
1.03
0.04
0.22

 

** on right indicates that 90% of subjects considered the map " good";
on left indicates that 90% of subjects considered the map "bad".
* on right indicates that 75% of subjects considered the map "good";
on left indicates that 75% of subjects considered the map "bad".

 

 

 

41

frequency good and bad shapes, and different categories did not have to be made,
I did not concern myself with the 75% good and bad shapes in the rest of the

analysis.

Index Values as Predictors of Perceptual Scores

Having described the perceptual results and found that there is consistency
among subject responses and that the shapes can be divided into categories of
acceptable and unacceptable amounts of distortion, the next step is to find the index
that is most closely related to the perceptual scores. That index can then be used to
develop a model for predicting perceptual shape distortion from the calculated index
values. The critical index values for acceptable and unacceptable distortion can also
be calculated. With such values cartographers could immediately deterrrrine the
acceptability of shapes on maps by comparing computed index values to the critical
values.

It is easy to see from the category Tables 4 and 5 that the only index values
that are related to the perceptual scores are the radial deviation index values. The
grid deviation values and the perimeter/ area ratio values vary widely within all of
the categories. The radial deviation values do not fit exactly into the perceptual
categories either. Especially obvious outliers are the low radial deviation values that
fall into high perceptual categories. Values for Greenland fit the best, while Africa’s
values are shuffled between the two most distorted categories.

If just the median value for the "bad" responses is used to separate the map

 

 

42

outlines into good and bad representations, the radial values fit quite well. The
radial deviation values have almost no overlap between these categories. The
exceptions in Table 4 are YUW for Australia, and YTE, RMC, VJZ, and UAM for
Africa. In Table 5 the exceptions are TIV, OFG, HXL, WFQ, and YUW.

To summarize the strength of the relationship between perceptual and radial
deviation values, Pearson and Spearman correlations were calculated. The
correlation coefficients for the separate and combined landmasses are shown in
Table 6. For completeness, the coefficients between perceptual scores and the other
two indices are shown as well.

As expected from visual inspection of the data in Tables 4 and 5, a high
degree of association exists between the radial deviation values and perceptual
scores. Values range from 0.625 to 0.805 for the radial index. Tire other two indices
have correlation values ranging from -0.008 to 0.544. Although only the -0.008 value
lacks significance, clearly, the radial deviation index best approximates the subjects’
perceptual scores.

The next step in the analysis is to mathematically define the relationship
between the perceptual scores and the index that best approximates the perceptual
scores. Figure 8 shows the relationship between the radial deviation index values and
the median perceptual scores for each landmass. The different categories are
represented by different point symbols: plus sign for category 1 (low perceptual
distortion); square for category 2; circle for category 3; x for category 4; and triangle

for category 5 (high perceptual distortion).

 

43

Table 6
Correlation coeflicients between perceptual scores
and distortion indices

AUSTRALIA NUMBER OF OBSERVATIONS: 528

Pearson Spearman P

Radial Deviation Index 0.669 0.689 0.(X)0
Grid Deviation Index 0.544 0.424 0.000
Perimeter/Area Index 0.141 0.130 0.001
GREENLAND NUMBER OF OBSERVATIONS: 528

Pearson Spearman P

Radial Deviation Index 0.805 0.766 0.000
Grid Deviation Index —0.008 0.105 0.856
Perimeter/Area Index 0.448 0.493 0.000
AFRICA NUMBER OF OBSERVATIONS: 528

Pearson Spearman P

Radial Deviation Index 0.625 0.650 0.000
Grid Deviation Index 0.309 0.300 0.000
Perimeter/Area Index 0.302 0.308 0.000
COMBINED NUMBER OF OBSERVATIONS: 1584

Pearson Spearman P

Radial Deviation Index 0.666 0.704 0.000
Grid Deviation Index 0.327 0.326 0.000
Perimeter/Area Index 0.209 0.293 0.000

 

 

 

 

 

 

44
AUS

m

2

§

'3

B

a.

D

8

8.

.9

B

E

0.0 0.2 0.4 0.6 0.8 1.0 1.2
radial deviation index values

GRN

{n

8

9

E

:3

‘5.

0

8

8.

E

'U

D

E

0.0 0.5 1.0 15 2.0
radial deviation index values

AFR

is”

8

m

‘8

5

‘5.

9

8.

.9

8

E

 

0.2 0.3 0.4 05 0.6 0.7
radial deviation index values

Figure 8. Plots of median perceptual scores with radial deviation index values.
+ =category 1 (low perceptual distortion); I =category 2; O =category 3;
x =category 4; V =category 5 (high perceptual distortion)

45

Since the relationship appears to be linear, linear regression was used to fit

a function to the data using the formula:

perceptual score = constant + coefficient ‘ radial deviation indexvalue

or

PSCORE = CONSTANT + b ‘ RDEV
Regression results for the separate landmass equations and a combined equation are
shown in Table 7.

Regression equations for the separate landmass equations have R-squared
values in the range of 0.4 to 0.6 and all three equations are significant (p < 0.001).
Values for the constants range from 1.3 to 2.0; the values for the slope of the line for
each landmass range from 6.2 to 14.9. The standard coefficient, the coefficient that
would result if regression was used on standardized values of the variables, remains
fairly stable (.63 to .81). All of the constants and slope values are statistically
significant.

Tire results from the combined regression analysis have an R-squared value
of 0.444 which indicates that over 44% of the variation is explained by the regression
equation. The significance level of the equation is high (p < 0.001) and the standard
coefficient is 0.67. These results parallel the landmass specific results and suggest
that one model for shape distortion is all that is needed for analyzing shape
distortion. If this combined equation was not as strong as the landmass specific
results, different models would be needed for different shapes on maps.

The final step in the analysis is to calculate the critical index values that

 

 

46

Table 7

Regression results for each landmass and combined landmasses
PSCORE = CONSTANT + b*RDEV

 

AUSTRALIA n = 528

R = 0.669 R-squared = 0.448 Adjusted R-squared = 0.447
standard error of estimate = 2.525

Variable Coefiicient Std. error Std coef T P(2 tail)
CONSTANT 0.502 0.225 0.000 2.234 0.026
RDEV 9.009 0.436 0.669 20.648 0.000
ANALYSIS OF VARIANCE

Source Sum-of—squares df Mean-square F-ratio P
REGRESSION 2717.871 1 2717.871 426.358 0.000
RESIDUAL 3353.054 526 6.375

GREENLAND n = 528

R = 0.805 R-squared = 0.648 Adjusted R-squared = 0.647
standard error of estimate = 1.987

Variable Coefficient Std. error Std ooef T P(2 tail)
CONSTANT 1.301 0.130 0.000 10.009 0.000
RDEV 6.247 0.201 0.805 31.086 0.000
ANALYSIS OF VARIANCE

Source Sum-of-squares df Mean-square F-rau'o P
REGRESSION 3815.602 1 3815.602 966.329 0.000
RESIDUAL 2076.940 526 3.949

AFRICA n = 528

R = 0.625 R-squared = 0.391 Adjusted R-squared = 0.390
standard error of estimate = 2.632

 

Variable Coefficient Std. error Std coef T P(2 tail)
CONSTANT -2.016 0.383 0.000 -5.264 0.000
RDEV 14.930 0.812 0.625 18377 0.000
ANALYSIS OF VARIANCE

Source Sum-of-squares df Meansquare F-ratio P
REGRESSION 2339.883 1 2339.883 337.717 0.000
RESIDUAL 3644.405 526 6.929

COMBINED r1 = 1584

R = 0.660 R-squared = 0.444 Adjusted R-squared = 0.444
standard error of estimate = 2.514

Variable Coefficient Std. error Std coef T P(2 tail)

CONSTANT 1.069 0.116 0.000 9.226 0.000

RDEV 7.496 0.211 0.666 35.552 0.000
ANALYSIS OF VARIANCE

Source Sum—of—squares df Mean-square F-ratio P

REGRESSION 7988.608 1 7988.608 1263.965 0.000

RESIDUAL 9998.677 1582 6.320

 

 

 

47

correspond to the critical perceptual category breaks. The perceptual values for the
combined landmass category breaks (average of the last value in one category and
the first value in the next) were entered into the combined equation to calculate a
radial index value for that break point. Table 8 lists the resulting critical index _

values.

Table 8
Predicted radial index category breaks

Perceptual Breaks Corresponding Radial Index Value
M

0.65
3.38
7.78

 

 

   
    
 
    

Interestingly enough, the first calculated value is negative. Negative index

values cannot exist because the radial index deviation is calculated using the root
mean square deviation. The model, however, is linear and can produce negative
values. Even though it was a "good" fit, the model does not function perfectly.
The corresponding index value for the median BAD perceptual score (3.34)
is 0.303. This value should divide the shapes into "good" and "bad" by their radial
index values. But, if we look again at the table of category breaks (Table 5), we see
that this index value of 0.303 divides the shapes differently than the perceptual
results suggest. Many more values would be considered "bad", but none of the
current "bad" outlines would be changed to "good". Visual inspection indicates that
an index value of 0.40 would do a better job of approximating the perceptual

categories. .

 

 

 

48
Sim—mar!

In summary, subjects as a group did not distinguish clearly the amount of
shape distortion on each outline. As a result, 1 divided the shapes into several
categories based on median distortion ratings and dispersion about the medians. The
median response to the question about the first "bad" representation of the true
shape was used to divide the outlines into those with acceptable and those with
unacceptable amounts of distortion. The shapes that were considered acceptable or
unacceptable 75% and 90% of the time were listed and found to fit into the
categories based on the median BAD value.

When corresponding index values were matched with the shapes in each
category, the only index that approximated the perceptual scores was the radial
deviation index. Its values fit almost perfectly into the categories, especially when
only the acceptable and unacceptable categories were considered. Correlation results
further demonstrated the strong relationship between the radial index values and the
perceptual scores.

Plots of the radial values and perceptual scores suggested that a linear
function would describe the relationship. The high degree of significance in results
from regression analysis showed that the fit was good. A linear model can be used
to predict perceptual amounts of shape distortion from mathematically calculated
shape distortion as measured by the radial index. When the model was used to find

index values for the perceptual category breaks, the results were lower than expected.

 

 

 

 

CHAPTER V

DISCUSSION AND SUGGESTIONS FOR FURTHER RESEARCH

While carrying out this research, I made several observations about the subject
testing procedure and the subjects’ results, the effectiveness of the radial deviation
index, and the model that was developed to predict perceptual shape distortion from
the calculated index values.

In the subject testing procedure a number of concerns were raised. First of
all, sixteen shapes are a challenge to rank and several subjects commented on there
being too many. I wanted to have enough to make regression analysis a reasonable
method to use, and even though I had many pairs of values (16 shapes times 33
subjects), the size of my shape sample was only sixteen per landmass. Results, then,
may be biased by the particular selection in my modest number of outlines. In
particular, the categories as determined by natural breaks among medians would be
affected by the specific sample of maps included in the test.

Constraining the subjects’ responses by scale endpoints may also have affected
the results. Outlines that were placed consistently near those endpoints had very
skewed perceptual score distributions making the overall score distributions for each
outline shape bimodal. The scores tended to cluster around the endpoints and
spread out more in between. Such a distribution makes regression results unreliable.

A different method of obtaining perceptual scores whereby subjects consider the

49

 

 

50

globe the zero endpoint and then mark the shapes by how much they differ from the
globe may have given different, and potentially more useful, results.

The radial deviation index is not perfect in its approximation of the perceptual
scores (refer to Figure 5). The graphs of the index values versus the perceptual
scores have linear trends, but the relationships are not perfectly linear. The category
symbols do not cluster perfectly either. Especially obvious are symbols that belong
to the third or fourth (high perceptual distortion) categories but have relatively low
index values. The shapes that produced these scores are all distorted similarly; they
are all long and thin (see YUW, Australia; UAM, Africa; and IJX, Greenland in
Appendix B). Apparently such shape change is highly noticeable to human subjects,
but the index value is only moderately high, resulting in an underestimation of
perceived shape distortion.

Overestimation also occurs. Symbols that represent low perceptual distortion
categories but have relatively high amounts of distortion are visible in Figure 5.
These include VON for Africa and MIH and KBG for Australia. In these outlines,
the distortion occurs in places where it is less visible because of the nature of the
continental shape itself. For example, in MIH and KBG of Australia, the distortion
just increased the bulge at the southeastern part of the landmass. It does not seem
to affect the overall look of the landmass to the degree predicted by the radial
deviation index. In VON, the distortion flattens the northern section of Africa, but
the overall shape of Africa changes little perceptually.

Because the overestimation and underestimation occur for similar shapes, the

51

index may be improved by adding an additional parameter to account for these
systematic errors. If such an improvement can be made, the value of the index as
a measure of shape distortion would increase.

Clearly, the radial index is not perfect and that means that the regression
equation is not a perfect model for perceptual shape distortion. This is apparent in
Table 8 which shows the index values that the model calculated for the category
breaks. The index values were very low, including a nonsensical negative value. A
different regression model that finds the path of standard deviations between the
variables may be a somewhat better model in that it is a compromise between a least
squares Y = a + bX model and the X = a + bY model. It would not eliminate the
possibility of calculating negative values, however.

Another observation about the index and the model is that they are only
meant to analyze one shape at a time. Cartographers would have to analyze each
shape on their maps to see if those shapes are acceptable representations. Then, if
some shapes are acceptable and others not, which is often the case on non-
interrupted maps, we would need to decide if the average amount of distortion
should be the measure of overall distortion on maps or if we should set a minimum
standard to be met by all shapes.

Another limitation of my study is that it does not give any suggestions as to
how distortions of shape should be weighed against other properties of projections
that are important to producing good maps nor does it tell us whether the use of the

index would give us more useful information about shape distortion than the

 

lufif.

 

.L.

 

52

traditional visual examination of the map as a whole. Equal area projections, for
example, often have severe shape. distortion, yet knowing the amount of each type
of distortion will not necessarily tell us which projection to select.

Despite these limitations, this research is an encouraging beginning to
analyzing shape distortion on maps. The simple radial deviation index goes a long
way toward automated shape distortion analysis. With further improvement, a
"smart“ computer selection routine such as the one proposed by Nyerges and

Jankowski (1989) could be more complete in its selection criteria.

Suggestions for Further Research

To see if the radial deviation index continues to be a good approximator of
perceptual distortion on maps, more shapes need to be tested. The category break
values and "bad" scores need to be further tested as well to see if they are reliable
limits for acceptable shape distortion on map projections. With such limits, standards
could be developed and map projections could be analyzed to see which ones
represent continental shapes acceptably. Different subject evaluation techniques can
be tried to see which ones provide the most usable results.

Most of all, tests of the usefulness of the index’s application will be in order
eventually to see if cartographers choose better map projections using quantitative
shape distortion information. Other potential applications, such as using this index
to develop new projections that rrrinirrrize shape distortion on maps, might also be

elaborated and tested.

 

 

 

CHAPTER VI

SUMMARY

Shape is an important part of map projection selection but no guidelines for
selecting projections with minimal shape distortion exist. This in not because
cartographers do not realize the importance of shape, but because shape is difficult
to define and measure.

Past attempts to measure shape have all had shortcomings. Some shape
indices give the same value to very different shapes, others are limited by their
complexity. More importantly, the shape measures were never tested to see if they
correspond to the way people see shape, and relatively limited attention has been
paid to adapting them to measure shape change.

My intent in conducting this research was to find and describe the relationship
between three shape distortion indices and perceptual shape distortion values as
judged by human subjects. The index which best approximated the perceptual values
was used to develop a model of perceptual shape distortion and establish guidelines
for selecting map projections when shape distortion is a concern.

To obtain perceptual values of shape distortion, a number of subjects
compared continental map outlines to their corresponding globe outlines and ordered

them from least distortion to most distortion. To record the order, the subjects

53

 

 

 

54

marked and labelled a ten-inch scale, spacing the markings to show the different

amounts of distortion in each outline. They were also asked to point out the first
shape that was a "bad" representation of the true continent’s shape. The subjects
responses could be divided into categories of acceptable and unacceptable amounts
of distortion. The median response to the question concerning the "bad"
representation was the dividing point.

The index that best approximated the perceptual scores was the radial
deviation index. To calculate this index, I measured radial distances from the
geometric center of map shapes to equidistant points on the shape’s outline. I then
calculated the deviation of those map distances from their corresponding globe
distances (in standard deviation units). The resulting root mean square deviation
value was the index of shape distortion.

These values also fit into the categories with a few exceptions. The index
value that appeared to divide the shapes into acceptable and non-acceptable was 0.4.
This means that shapes with a radial index value less than 0.4 have acceptable
amounts of shape distortion while shapes with index values greater than 0.4 are
generally unacceptable representations of the true shape on the globe.

The regression equation that modelled the relationship between the
perceptual scores and the index values was highly significant(p < .001) further
emphasizing the strength of the relationship between the radial index values and
subjects’ perceptual values. When the equation was used to calculate index values

for the perceptual category breaks (they were only visually approximated before), the

 

 

 

55
result indicated that an index value of 0.303 should be used to divide the shapes into

acceptable and unacceptable. Few shapes would be acceptable with this value. The
visually assigned value of 0.4 seems more reasonable.

The regression equation is not perfect. The perceptual scores were not
normally distributed making the equation less than reliable. Overestimation of
perceptual scores occurred when distortion occurred in limited parts of the shape.
A shape can sometimes be distorted and yet maintain its overall appearance.
Underestimation occurred when a shape was distorted in such a way that it became
long and thin. Such distortion was apparently more noticeable to subjects than other
distortion.

Overall, the results of this study do give some new insight into the perception
of shape distortion and how perceptual shape distortion can and cannot be measured.
Cartographers can now begin to apply this model and further test it to see which
projections best preserve shape and to see if the model chooses the same projections
as are currently considered the best when shape is of concern. The category breaks
may also be used to deterrrrine whether continental shapes are adequate or
inadequate representations of their true globe shapes.

More work needs to be done on this topic of shape distortion. Many more
shape indices exist and could be tested. Methods for summing the amounts of
continental distortion on projections need to be developed so that cartographers do

not have to analyze continents separately as I have done in this study. More shapes

 

 

 

56

need to be studied to test the consistency of the category breaks and the coefficients

in the regression model. The research I have done is just a beginning to a very

complex problem, but the results are encouraging and should be explored further.

 

 

 

 

Appendix A

Programs for Calculating Shape Distortion Index Values

57

’PROGRAM FOR CALCULATING THE PERIMETER OF AN OBJECT ON A
’SPHERE

’This program accepts radian data that has been printed from ’the prorad qb program
and calculates the perimeter of that ’figure. The first and last coordinates in the data
are the ’same; so, the entire perimeter is calculated.

INPUT "Enter input filename: "; infileS
OPEN infile$ FOR INPUT AS #1
PRINT "Program is running please wait."

’define variables and constants

DEFDBL GD, 1., Z

DEFINT K

CONST pi = 3.14159

CONST radius = (6371100 ' 100) / (4E+07 ' 2.54)
’radius of generating globe in inches

LET dsum = 0

’read in data pts., calculate distances between them and sum ’to get the perimeter

INPUT #1, lngl, latl, k1
20 DO UNTTL EOF(1)
INPUT #1, lng2, latZ, k2
LET cosz = SIN(lat1) ' SIN(lat2) + COS(lat1)' COS(iat2) ‘COS(lng1-lng2)
IFcosz=1TI-IEN2 = 0
’when cosz=1 the SQR function in the next step is undefined
ELSE
LET z = (pi / 2) - ATN(cosz / SQR(1 - cosz " 2))
END IF
LET d = radius ’ z
LET dsum = dsum + d

LET lngl = IngZ

LET latl = lat2

LET k1 = k2
LOOP

PRINT "The perimeter in inches is: "; dsum
END

58
’PROGRAM FOR CALCULATING THE PERIMETER OF AN OBJECT ON A
’PLANE

’This program accepts WORLD postscripts files once the last ’extra lines are stripped
off in a word processing program ’and calculates the perimeter of of the figure using
the x,y ’coordinates. The first and last coordinates in the data ’are the same; so, the
entire perimeter is calculated.

INPUT "Enter input filename: "; infileS
OPEN infileS FOR INPUT AS #1
PRINT "Program is running please wait."

’define variables and constants
DEFDBL D, X-Y

DEFSTR K

LET dsum = 0

’read past first 12 lines
FORi = 1 TO 12

LINE INPUT #1, header$
NEXT i

’read in data pts., calculate distances between them and sum ’to get the perimeter

INPUT #1, x1, yl, k
DO UNTIL EOF(1)
INPUT #1, x2, y2, k
IFx2 = 5.5 AND y2 = 4.25 THEN
GOTO 50
ELSE
LETd = SQR((xl-x2)"2 + (yl-y2)"2)
LET dsum = dsum + d

LETxl = x2
LETy1= y2
ENDIF

LOOP
50 PRINT "The perimeter in inches is: "; dsum
END

 

 

 

59
’PROGRAM FOR CALCULATING THE AREA OF AN OBJECT ON A PLANE

’This program accepts WORLD postscripts files

’and calculates the area of of the figure using the x,y ’coordinates. The first and last
coordinates

’in the data are the same; so, the entire area is ’calculated.

’It uses the SCALE of the map plane to calculate and print

’the area of the object on the earth.

CLS

INPUT "Enter .per input filename: "; infile$
OPEN infileS FOR INPUT AS #1

PRINT "Program is running please wait."

’define variables and constants

DEFDBL D, G, M, X-Y

DEFSNG A

DEFSTR K

LET msum = 0

LET scale = 6.3E+07 ’SCALE OF MAP OUTLINES’

’read past first 12 lines
FOR i = 1 TO 12

LINE INPUT #1, header$
NEXT i

’read in data pts. and begin area calculation
INPUT #1, x1, yl, k
DO UNTIL EOF( 1)
INPUT #1, x2, y2, k
IF x2 = 5.5 AND y2 = 4.25 THEN
GOTO 50
ELSE
LETm = (x2‘y1)-(x1‘y2)
LET msum = msum + m

LETx1= x2
LETy1= y2
END IF

LOOP

50 LET area = .5 ‘ ABS(msum) ’area in sq.inches ON MAP’
LPRINT "The area in MAP inches for: "; infileS; area

END

 

 

 

60
’PROGRAM FOR CALCULATING GLOBE GRID DISTANCES

’This program uses radian data from prorad and calculates ’globe distances between
grid coordinates. The file must have ’data in order from left to right across the
columns and then ’down the rows. The globe distances are printed to a gd file
’Three constants must be specified in the program: the scale ’of the generating globe,
and the number of rows and columns ’in the data.

PRINT "Use radian data for input."

INPUT "Enter ...grid.rad file for input: "; infile$

OPEN infileS FOR INPUT AS #1

INPUT "Enter gd... output filename: "; outfile$

OPEN outfileS FOR OUTPUT AS #2

PRINT "Deviation index program is running. Please wait."

’definition of variables and constants
DEFINT P-Q
DEFDBL A, OD, L-M
CONST pi = 3.14159
CONST radius = (6371100 ‘ 100) / (1E+08 ‘ 2.54)
’radius of generating globe in inches
’1e+08 is the map scale for AFR, 4e+07 for GRN, 6.3e+07 for ’AUS

LET p = 17 ’# of rows in data grid
LET q = 16 ’# of columns in data grid

’GRN: p =
’AUS: p = 7
’AFR: p = 1

DIM lat(p, q). lng(P. <1)

’read in grid coordinate array
FOR i = 1 TO p
FOR j = 1 TO q
INPUT #1, lng(i, j), lat(i, j), a
NEXT j
NEXT i

’calculate distances across array, i.e., along parallels
FOR 1 = 1 TO p
FOR j = 1 TO q - 1
d2 = radius ’ COS(lat(i,j)) ‘ (lng(i,j+1) - lng(i,j))
PRINT #2, d2, "0"

 

 

61

NEXTj
NEXTi

’calculate distances down array, i.e., along the meridians
FOR j = 1 TO q
FOR i = 1 TO p - 1
d2 = radius ‘ (lat(i, j) - lat(i + 1, j))
PRINT #2, d2, "1"
NEXT i
NEXT j
CLOSE #1
CLOSE #2
PRINT "Session finished. Spherical grid distances saved to gd file"
END

62

’PROGRAM FOR CALCULATING MAP GRID DISTANCES AND THE GRID
INDEX ’VALUES

’This program uses WORIJ) postscript files and calculates map

’distances between grid coordinates. The postscript file must ’have data in order
from left to right across the columns and ’then down the rows. The mean deviation
of the map grid ’distances from the globe distances in a gd... file is printed ’as the
deviation index.

’THE NUMBER OF ROWS AND CLOUMNS IN THE DATA MUST BE
SPECIFIED

CLS

PRINT "This program calculates the mean linear deviation index."
PRINT "Use ...grd files for input."

INPUT "Enter input .grd filename (postscript file): "; infile$
OPEN infileS FOR INPUT AS #1

INPUT "Enter the gd... filename for input: "; infile1$

OPEN infilelS FOR INPUT AS #2

PRINT "Deviation index program is running. Please wait."

’definition of variables and constants
DEFINT F, N, P—Q
DEFDBL D, X-Y
DEFSNG A, C, L-M
LET numberh = 0

LET numberv = 0
LET dewsum = 0
LET devhsum = 0
LET p = 7 ’# of rows in data grid

LET q = 10 ’# of columns in data grid
DIM KG), (1). 1'0). <1)

’read past first lines
FOR i = 1 TO 12
LINE INPUT #1, header$
NEXT i
’read in grid coordinate array
FOR i = 1 TO p
FOR j = 1 TO q
INPUT #1, x(i, j), y(i, j), a
NEXT j
NEXT i

’calculate distances across array, i.e., along parallels
FOR i = 1 TO p

 

 

 

 

63

FOR j = 1 TO q - 1
d1 = SQR((X(iJ)- 1((i.i+1))"2 + (y(iJ)- y(i.i + 1))“2)
INPUT #2, d2, flag
IF flag = 1 THEN
PRINT "ERROR--distances do not match up"
GOTO 100
END IF

devh = (d1 - d2) A 2
devhsum = devhsum + devh
numberh = numberh + 1
NEXT j
NEXT 1

’calculate distances down array, i.e., along the meridians
FOR j = 1 TO q
FOR i = 1 TO p - 1
d1 = SQR((X(iJ)- X(i+ 1.1))22 +(y(i.i)-y(i+1,j))22)
INPUT #2, d2, flag
IF flag = 0 THEN
PRINT "ERROR--distances do not match up"
GOTO 100
END IF
dew: (d1-d2)"2
devvsum = devvsum + dew

numberv = numberv + 1
NEXT i

NEXTj

’calculate the mean linear deviation
LET n = numberh + numberv
LET devsum = dewsum + devhsum
meandev = devsum / n

LPRINT ”grid deviation index "; infileS, meandev
CLOSE #1
CLOSE #2

PRINT "Session finished. "

100 END

 

 

 

64
’PROGRAM FOR RADIAL DEVIATION ON THE GLOBE

’This program accepts radian eq data in prorad format. It ’calculates the length of
radii from the center to the other ’points in the data file. The center coordinates

must be the ’first value in the data file The mean radius of the actual ’globe distance
is calculated and each radial distance is ’tumed into deviation units from the mean
(z-scores) which ’are written to an output file.

INPUT "Enter ...eq.rad filename for input: "; infileS
OPEN infile$ FOR INPUT AS #1
INPUT "Enter the rd... filename you want as output: "; outfile$

OPEN outfileS FOR OUTPUT AS #2
PRINT "Program is running please wait."

’define variables and constants

DEFDBL C, K-M, R-S, Z

CONST pi = 3.14159

CONST radius = (6371100 ' 100) / (1E+08 ‘ 2.54)
’radius of generating globe in inches

LET n = 97 ’the number of equidistant data points

DIM d2(n)

DIM dev2(n)

DIM 22(n)

LET sumd2 = 0

LET sumdev2 = 0

LET sumdevz = 0

’read in center coords, the first data values in the radian data ’file
INPUT #1, clong, clat, k1

’read in data pts. one at a time, calculate spherical
’distances and sum. Sum needed to find mean radius.
FOR i = 1 TO 11
INPUT #1, lng, lat, k1
LET cosz = SIN(lat)‘SIN(clat) + COS(lat)‘COS(clat)‘COS(lng-clong)
chosz = 1THEN LETz = 0
’when cosz = 1, the SQR function in the next step will be undefined
ELSE
LET z = (pi / 2) - ATN(cosz / SQR(1 - cosz " 2))
END IF
LET d2(i) = radius ‘ z
LET sumd2 = sumd2 + d2(i)
NEXT 1
’calculate mean radii

 

 

 

65

LET meand2 = sumd2 / n
’calculate deviation from mean, put in standard deviation ’units, z-scores
FOR i = 1 TO 11
LET dev2(i) = d2(i) - meand2
LET sumdev2 = sumdev2 + (dev2(i)) " 2
NEXT i
LET stdev2 = SQR(sumdev2 / n)
FOR i = 1 TO 11
LET 22(i) = dev2(i) / stdev2
PRINT #2, 22(i)
NEXT i
CLOSE #1
CLOSE #2
PRINT "Program finished. File of z-scores saved."
END

 

 

 

66
’PROGRAM FOR RADIAL DISTORTION MEASURE

’THE NUMBER OF EQUIDISTANT POINTS MUST BE SPECIFIED

’This program accepts WORLD postscript files. It calculates

’the length of radii from the center to the other points in ’the data file. The center
coordinates must be the first ’value in the data file The mean radius of the map
distance

’is calculated and each radial distance is turned into ’deviation units from the mean.
The mean deviations of the ’map are compared to the mean deviations on the globe
which ’are in an rd data file, and the root mean square error is ’calculated and used
as the shape index.

CLS

INPUT "Enter the .rdv file you want as input: "; infileS
OPEN infileS FOR INPUT AS #1

PRINT "Enter the rd... deviation file for the corresponding"
INPUT "country: "; infilelS

OPEN infilelS FOR INPUT AS #2

PRINT "Program is waning please wait."

’define variables and constants
DEFDBL C, K-M, R-S, X-Z
DEFSNG I
LET n = 94 ’the number of equidistant data points
DIM d1(n)
DIM dev1(n)
DIM zl(n)
DIM 22(n)

LET sumdl = 0
LET sumdevl = 0
LET sumdevz = 0

’read past first lines
FOR 1 = 1 TO 12
LINE INPUT #1, headerS
NEXT 1

’read in center coords, the first data values in the outpost ’file
INPUT #1, cx, cy, k

’read in data pts. one at a time, calculate planar
’distances and the squared difference and sum.
FOR i = 1 TO n

 

 

67
INPUT #1, x, y, k

LET d1(i) = SQR((x-cx) *2 + (y-cy) .2)

LET sumdl = sumdl + d1(i)
NEXT i
PRINT x, y, "last x,y"

’calculate mean radii
LET meandl = sumdl / n
PRINT meandl, "mean"
’calculate deviation from mean, put in standard deviation ’units, z-scores then
calculate the deviation of the map ’z-scores from the globe z-scores
FOR i = 1 TO 11
LET dev1(i) = d1(i) - meandl
LET sumdevl = sumdevl + (dev1(i)) A 2
NEXT i
LET stddevl = SQR(sumdevl / n)
PRINT stddevl, "std. dev."
’change to standard deviation units and calculate the root ’mean square error
FOR i = 1 TO n
INPUT #2, 22(1)
LET zl(i) = dev1(i) / stddevl

LET sumdevz = sumdevz + (zl(i) - 22(i)) " 2
NEXT 1
LET index = SQR(sumdev2 / 11)

PRINT infileS; " radial index", index
CLOSE #1
CLOSE #2

END

 

 

 

Appendix B

Materials for Subject Testing

 

 

68
Appendix B

All of the materials used for the subject testing procedure except for the globe
are included. The script was used with every subject and consent forms were signed
by every subject before the testing began. Each subject was given a pencil and
recording sheet after ordering the shapes from least to most distortion. Subjects
were permitted to use as much time as needed to complete the task (normally one-
half hour total).

The record sheet is reduced to 80% but is shown as it was given to subjects.
I labelled each line for the landmass shapes that were to be marked on it. Because
the order of the landmass evaluations rotated for each subject, the line labels also
changed so that the top line on the record sheet was always labelled with the first
landmass to be evaluated.

The shape outlines are shown at 70% actual size. They were cut apart and

placed on one-quarter page cardstock for the subject evaluation.

 

 

 

69
SCRIPT

Hi. Thanks for helping me with my research. Before you begin I need you to read
and sign this consent form. [Students read and sign]

You will be evaluating three landmasses for shape distortion. [Three stacks of
shapes are shown] Each stack contains sixteen outlines of one landmass, all with
varying amounts of shape distortion. The first thing I want you to do is put the
sixteen outlines in order from least amount of shape distortion on your left, to most
on your right. The globe shows the true shape so compare each shape to the globe.
You’re only concerned with the overall amount of shape distortion. Size is not a
factor. Just because one shape is bigger, doesn’t mean that it has more distortion.
When you get them in order let me know and I’ll explain how to use the record
sheet. There is no correct order. It’s what you think is correct that matters. You
have as much time as you need. [Wait until they’ve finished with this task]

Now to record your order, you’ll label the left tick mark on the record sheet with the
letters on the shape with the least amount of distortion, and the right tick mark with
the letters on the shape having the most distortion. Then, you will be marking and
labelling the line for each shape to indicate how much distortion you see. So, if two
shapes have similar amounts of distortion, the marks will be close together, and if the
shapes vary more greatly in their amount of distortion, the marks will be farther
apart. The hard part is to get your spacing correct so that the marks all fit between
the end points. I can suggest that you pretend the table is the scale. put the least
and most distorted shapes at the edges and arrange the other shapes leaving spaces
or putting them close together until you get a spacing you like. Then mark Off on
the paper to look like your arrangement on the table. You may also want to find the
shape with a medium amount of distortion and mark it near the rrridpoint of the scale
and then work with each half. [Wait again]

One last part. As you look at each shape starting with the one having the least
distortion, which is the first outline that you would consider to be a bad
representation of that landmass. Remember to check with the globe. Use an arrow
to point to the label of the landmass you chose on your record sheet.

Now just repeat that process for the other two landmasses.

Thank you, that’s all.

 

 

 

70
CONSENT FORM

1. I freely and voluntarily consent to take part in a scientific study being conducted
by Dawn Carlson by completing and returning this questionnaire.

2. I understand that the study investigates shape distortion on map projections. The
study has been explained to me, and I understand the explanation that has been
given, and what my participation will involve.

3. I understand that the study involves only one session that will take approximately
one half hour, but I may take as much time as I need to finish the session.

4. I understand that I am free to discontinue my participation at any time during
the procedure without penalty.

5. I understand that the results of my participation in the study will be kept in strict
confidence, as will those of all other individuals participating. In other words, all
participants will remain anonymous in the report of results.

6. I understand that, at my request, I can receive additional explanation of the study
after my participation is completed by contacting Dawn.

Signed: _________ Date: _

 

 

 

 

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REFERENCES

 

 

 

REFERENCES

Beyer, William H. 1984. CRC Standard Mathematical Tables, 27th Ed. Boca Raton,
Florida: CRC Press, Inc.

Bookstein, Fred L. et. al. 1985. Momhometries in Evolutionary Biology Special
Publication 15, The Academy of Natural Sciences of Philadelphia.

Boyce, Ronald R. and W.A.V. Clark. 1964. 'The Concept of Shape in Geography,"
The Geographical Review 54:561-572.

Canters, Frank. 1989. "New Projections for World Maps/ A Quantitative-Perceptive
Approach," Cartographica 26(2):52-71.

Dent, Borden D. 1987. "Continental Shapes on World Projections: The Design of a

Poly-Centered Oblique Orthographic World Projection," The Cartographic
Journal 24(2):117-124.

Espenshade, Edward B., Jr. ed. 1986. Goode’s World Atlas Chicago, Rand McNally,
p. x.

 

Frolov, Yu. S. 1975. "Measuring the Shape of Geographical Phenomena: A history
of the Issue," Soviet Goegraphy Review and Translation 16(10):676-687.

Garver, John B., Jr. 1988. "New Perspective on the World," National Geographic
Magazine, 174(8): 910-913.

Lee, David R. and G. Thomas Sallee. 1970. "A Method of Measuring Shape,"m
Geographical Review 60:555-563.

Mahling, DH. 1973. Coordinate Systems and Map Projections, London: George
Philip and Son Ltd.

Muehrcke, Phillip C. 1978. Map Use: Reading, Analysis, and Interpretation. JR
Publications: Madison, WI.

Nyerges, Timothy L. and Piotr Jankowski. 1989. "A Knowledge Base for Map
Projection Selection," The American Cartographer 16(1): 29-38.

Pavlidis, Theodosios. 1978. "A Review of Algorithms for Shape Analysis," Computer
Graphics and Image Processing 7:243-358.

85

 

 

 

REFERENCES

Beyer, William H. 1984. CRC Standard Mathematical Tables 27th Ed. Boca Raton,
Florida: CRC Press, Inc.

 

Bookstein, Fred L. et. al. 1985. Morphometrios in Evolutionag Biology Special
Publication 15, The Academy of Natural Sciences of Philadelphia.

Boyce, Ronald R. and W.A.V. Clark. 1964. "The Concept of Shape in Geography,"
The Geographical Review 54:561-572.

Canters, Frank. 1989. "New Projections for World Maps/ A Quantitative-Perceptive
Approach," Cartographica 26(2):52-71.

Dent, Borden D. 1987. "Continental Shapes on World Projections: The Design of a

Poly-Centered Oblique Orthographic World Projection," The Cartographic
Journal 24(2):117-124.

Espenshade, Edward B., Jr. ed. 1986. Goode’s World Atlas Chicago, Rand McNally,
p. x.

 

Frolov, Yu. S. 1975. "Measuring the Shape of Geographical Phenomena: A history
of the Issue," Soviet Goegraphy Review and Translation 16(10):676-687.

Garver, John B., Jr. 1988. "New Perspective on the World," National Geographic
Magazine, 174(8): 910-913.

Lee, David R. and G. Thomas Sallee. 1970. "A Method of Measuring Shape,"_T_fi
Geographical Review 60:555-563.

Mahling, DH. 1973. Coordinate Systems and Map Projections, London: George
Philip and Son Ltd.

Muehrcke, Phillip C. 1978. Map Use: Reading, Analysis, and Interpretation. JR
Publications: Madison, WI.

Nyerges, Timothy L. and Piotr Jankowski. 1989. "A Knowledge Base for Map
Projection Selection," The American Cartographer 16(1): 29-38.

Pavlidis, Theodosios. 1978. "A Review of Algorithms for Shape Analysis," Computer
Graphics and Image Processing 7:243-358.

85

 

 

 

86

Peters, Aribert. 1984. "Distance-related Maps, "The American Cartographer 11(2):
119-131.

Robinson, Arthur H. and John P. Snyder. 1991. "Matching the Map Projection to the
Need," American Cartographic Association Special Publication Number 3.
American Congress on Surveying and Mapping. Bathesda, MD.

Sirnons, Peter L. 1974. "Measuring the Shape Distortion of Retail Market Areas,"
Geographical Analysis 6:331-340.

Snyder, JP. 1985. "Computer-assisted map projection research," U.S. Geological
Survey Bulletin 1629, United States Government Printing Office, Washington.

Snyder, John P. 1987. "Map Projections - A Working Manual," U.S. Geological

Survey Professional Paper 1395, United States Government Printing Office,
Washington.

Tobler, Waldo R. 1986. "Measuring the Similarity of Map Projection" The American
Cartographer 13(2): 135-139.

Voxland, Philip M. 1987. WORLD Projection and Mapping Program Reference
Manual Version 4.01. Social Science Research Facilities Center, University

of Minnesota.

 

Waterman, Stanley. 1984. "A Quantitative-Comparative Approach to Analysis of
Distortion in Mental Maps," The Professional Geographer 36(3):326-327.

 

 

 

 

 

 

 

 

 

 

 

 

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