Minimizing Map Distortion Using Oblique Projections
Thesis
Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in
the Graduate School of The Ohio State University
By
Jiaqi Zhang
Graduate Program in Geodetic Science
The Ohio State University
2017
Thesis Committee
Dr. Alan John Saalfeld, Advisor
Dr. C.K. Shum
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Copyrighted by
Jiaqi Zhang
2017
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Abstract
We provide a precise mathematical definition of map distortion, and we introduce several map projections that have minimal distortion for certain specific regions of the
Earth (namely, regions whose boundary consists of one or two parallels of latitude). The minimizing projections for these regions will all have normal aspect. We show that oblique aspect projections have the same distortion properties as their normal aspect counterparts on earth regions bounded by one lesser circle or by two lesser circles lying in parallel planes.
We summarize the good consequences of having small distortion, and we outline a strategy for finding the best oblique map projection of least distortion for any region on the Earth. We then implement that strategy to find projections of least distortion for some unusually shaped countries, states and other regions.
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Acknowledgments
It’s a great honor to have the opportunity to study and research at the Ohio State
University and these years are the most precious time of my life. I would like to thank my advisor Dr. Alan John Saalfeld, the most important and kind person in my academic life here. Besides instructing me knowledge and the way to conduct research, he cultivates my capacity for independent thinking and attitude on scientific research. Also, my thanks go to Dr. Michael G Bevis, who inspires me a lot and supports our research. Moreover, I want to show my respect and appreciations to those excellent professors in my department like Dr. Jekeli and Dr. Schaffrin. Finally, I would like to thank my parents for supporting me financially and mentally.
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Vita
2010 - 2014 B.S., Geographic Information System and Cartography, Wuhan University,
China
2014 – present M.S., Geodetic Science, School of Earth Sciences, The Ohio State
University
Fields of Study
Major Field: Geodetic Science
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Table of Contents
Abstract ...... ii Acknowledgments...... iii Vita ...... iv List of Tables ...... vii List of Figures ...... viii Chapter 1. Introduction ...... 1 1.1 Scale and Distortion ...... 1 Chapter 2. Background and Knowledge ...... 4 2.1 Projections...... 4 2.2 Scales Calculation ...... 7 2.3 Normal Aspect Projections ...... 10 2.3.1 Normal Aspect Azimuthal Projections ...... 11 2.3.2 Normal Aspect Cylindrical Projections ...... 11 2.3.3 Normal Aspect Conic Projection ...... 12 2.4 Oblique Projections ...... 13 2.5 Comparison of Normal and Oblique Aspect Projections ...... 21 2.6 Normal and Oblique Aspect Projection’s Equivalent Properties of Distortion ...... 23 Chapter 3. Oblique Azimuthal Projection ...... 25 3.1 Methodology ...... 25 3.2 Experiments ...... 28 3.2.1 Data Preprocessing...... 28 3.2.2 Main Programs ...... 31 3.3 Results Analysis ...... 32 Chapter 4. Oblique Cylindrical Projection...... 35 4.1 Methodology ...... 35 v
4.2 Experiments ...... 37 4.2.1 Data Preprocessing...... 37 4.2.2 Main Programs ...... 38 4.3 Results Analysis ...... 40 Chapter 5. Oblique Conic Projection ...... 43 5.1 Methodology ...... 43 5.2 Experiments ...... 45 5.2.1 Data Preprocessing...... 45 5.2.2 Main Programs ...... 45 5.3 Results Analysis ...... 46 Chapter 6. Evaluation...... 48 6.1 Evaluation of Performance ...... 48 6.2 Evaluation of Effectiveness ...... 50 Chapter 7. Conclusions ...... 56 7.1 Discussion on angle and area distortion...... 56 7.2 Practical significance of oblique projections ...... 58 Bibliography ...... 60
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List of Tables
Table 1. Classification of normal aspect tangent projections...... 10
Table 2. Distortion for the equidistant azimuthal projection of a circular area...... 14
Table 3. Distortion for cylindrical projections of band around a great circle...... 16
Table 4. Distortion for conic projections of band around a lesser circle...... 18
Table 5. Distortion for conic projections of band around a lesser circle...... 19
Table 6. Comparison of normal aspect projection VS. oblique aspect...... 21
Table 7. Distortions under different equidistant oblique projections...... 48
Table 8. US states fitting within a spherical cap...... 58
Table 9. Countries fitting within a spherical cap...... 59
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List of Figures
Figure 1. Geodesic curves on the datum surface and the projection surface...... 3
Figure 2. Map projection scale determination by curve tangents...... 4
Figure 3. Maps of Ohio using different projections (cylindrical on left, conic on right). .. 6
Figure 4. Maps of South America using different projections...... 7
Figure 5. Normal aspect projections' scales...... 8
Figure 6. Equidistant normal aspect projections' scales...... 9
Figure 7. Distortion = ((x/R)/sin(x/R)-1)*100...... 15
Figure 8. Distortion = ([2tan(x/2R)/sin(x/R)]-1)*100...... 15
Figure 9. Distortion of a given width band region (Unit of x: KM)...... 17
Figure 10. Distortion of a given width band region (Unit of x: KM)...... 19
Figure 11. Distortion of a given width band region (Unit of x: KM)...... 20
Figure 12. Normal VS. Oblique aspect azimuthal projected map...... 21
Figure 13. Normal VS. Oblique aspect cylindrical projected map...... 22
Figure 14. Normal VS. Oblique aspect conic projected map...... 23
Figure 15. Projection's property of distortion...... 24
Figure 16. Oblique azimuthal projections...... 25
Figure 17. Minimum fitting circle in 2D plane...... 26
Figure 18. Dot products of directional vector and boundary point vector...... 28 viii
Figure 19. Basic information of input data...... 29
Figure 20. Extract feature points from the simplified boundary of China...... 30
Figure 21. Running time results of oblique azimuthal projection by calculating dot product...... 33
Figure 22. Oblique cylindrical projection...... 35
Figure 23. Dot products of directional vector and boundary point vector...... 36
Figure 24. Extract feature points from the simplified boundary of Chile...... 38
Figure 25. Running results of oblique cylindrical projection by calculating dot product. 40
Figure 26. Running results of oblique cylindrical projection by calculating distortion. .. 41
Figure 27. Oblique conic projection...... 43
Figure 28. Extracted feature points from the simplified boundary of Italy...... 45
Figure 29. Running results of oblique conic projection by calculating distortion...... 47
Figure 30. Heat maps of distortion in China under different projections ...... 51
Figure 31. Heat maps of distortion in Chile under different projections...... 52
Figure 32. Heat maps of distortion in Italy under different projections...... 53
Figure 33. Heat maps of distortion in Vietnam under different projections...... 54
Figure 34. Heat maps of distortion in Greenland under different projections...... 55
Figure 35. Tissot indicatrix and map projections...... 56
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Chapter 1. Introduction
1.1 Scale and Distortion
The common notion of a map’s scale is the ratio of distances on the map to distances on the ground. This layman’s definition of “map scale as a single number” falls apart under closer mathematical scrutiny. This seemingly intuitive single number belies the fact there can never be a single ratio that works for all inter-point distances on a flat map depicting a region on round surface. There are always instead a range of scales; and if that range has both a non-zero minimum scale and a finite maximum scale1, then we define the distortion to be the percentage increase in going from the minimum scale to the maximum scale.
If min is the minimum scale and max is the maximum scale, then
distortion = [(max - min)/min] × 100%.
The fact that min cannot equal max (and hence min < max) shows that distortion is always bigger than 0. In order to analyze distortion properly, we must carefully choose our definitions of map projections and scale in order to determine the maximum scale and the minimum scale of a projection. One observation about distortion is very important.
Distortion does not depend on the size of the map in the following sense: If we photographically enlarge a map to make it X times higher and X times wider, called
1 Any representation of the whole earth in the plane has arbitrarily large scale, and hence no σmax. 1 uniform scaling by X, then all scales in the new enlarged map will be X times larger. So the new map’s minimum scale will be X times min and the new map’s maximum scale will X times max; and the ratio of those two extreme scales will be unchanged. So the enlarged map will have the same distortion as the original map. Since uniform scaling does not change distortion, we may assume that our minimum scale is 1.
In terms of collocating our projection surface in 3D space with our datum surface, the projection surface will always be tangent to the datum surface: The cylindrical projection surface will always be tangent at some great circle, the conic projection surface will always be tangent to the sphere along some lesser circle, and the azimuthal projection plane will be tangent to the sphere at a single point.
Regions that can be bounded by a not-too-large circle can be mapped by an oblique azimuthal projection with very little distortion.
A distortion of 1% or less does not have any discernible length distortion: all inter-point distances on the map are at most 1% more than their corresponding distances on the sphere datum surface.
What does it mean to have X% distortion? By uniform scaling, we can make the minimum scale equal to 1 and the maximum scale equal to (1 + (X/100)).
Distances on the sphere and in the plane are measured along geodesic curves. As Figure
1 illustrates, on the sphere, a geodesic curve between two non-antipodal points p and q is the minor arc (pq) of the great circle passing through them. In the plane, the geodesic curve between any two points p’ and q’ is the straight-line segment [p’q’] connecting them.
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q (pq) q=q’ -1 (pq) [p’q’] [pq]
p p=p’
-1
Figure 1. Geodesic curves on the datum surface and the projection surface.
Under a map projection, geodesics in general do not map onto geodesics. Suppose, nevertheless that under , the points p and q go to points p’ and q’, respectively.
There are 4 curves of interest now, the two geodesics (pq) and [p’q’] and their images under and -1, respectively, namely (pq) and -1[p’q’]. If we use ‖ ‖ to represent the length of a curve, and if has scales between 1 and (1+), where = (X/100), then since augments length (or at least is length-non-shortening) and since geodesics in each space have shorter length than any other connecting curve, we must have:
‖ (pq) ‖ ≤ ‖ -1[p’q’] ‖ ≤ ‖ [p’q’] ‖ ≤ ‖ (pq) ‖ ≤ (1+)‖ (pq) ‖
‖ (pq) ‖ ≤ ‖ -1[p’q’] ‖ holds because (pq) is a geodesic on the sphere.
‖ -1[p’q’] ‖ ≤ ‖ [p’q’] ‖ holds because is non-shortening; and hence, -1 is non- lengthening.
‖ [p’q’] ‖ ≤ ‖ (pq) ‖ holds because [p’q’] is a geodesic in the plane.
‖ (pq) ‖ ≤ (1+)‖ (pq) ‖ holds because is not too augmenting. 3
Chapter 2. Background and Knowledge
2.1 Projections
Let R, a region, be some subset of points on a sphere having non-empty interior. Then a map projection Π of R is a differentiable injective function Π from R to 2 such that Π has a differentiable inverse Π-1 defined on the image set Π(R).
Figure 2. Map projection scale determination by curve tangents.
A projection Π maps a region of the sphere (or other datum surface) to a flat (or developable) projection surface. Any curve α:[t0, t1] → R on the datum surface is transformed under the projection to a curve Π◦α:[t0, t1] → Π(R) on the projection 4 surface. The velocity tangent vector α’(t) to the curve on the datum surface has a corresponding velocity tangent vector (Π◦α)’(t) = d[(Π◦α)(t)]/dt to the transformed curve on the projection surface. The curve speeds at α(t) and (Π◦α)(t) are just the norms of the tangent vectors: ‖α’(t)‖ for the datum surface and ‖d[(Π◦α)(t)]/dt‖ for the projection surface.
The scale at any point α(t) on the curve in the velocity tangent direction α’(t) is the ratio of the two speeds: ‖d[(Π◦α)(t)]/dt‖/‖α’(t)‖. From differential geometry, we know that this ratio does not depend on our choice of curve α. Any other curve passing through the point and having the same tangent vector direction will produce the same ratio. For any point there are scales in every direction; and we can always construct a curve passing through the point in any given direction and use that curve to find the scale. At any point there are four scales of special interest. It is common practice to use the letters h and k to represent scales along the meridian and parallel curves, respectively. In other words, h denotes the scale in the north-south directions, and k denotes the scale in the east-west directions. Since at any point there is a scale in every direction, it is conventional to use the letters a and b to denote the maximum and minimum scales at the point, respectively.
For a large group of projections called Normal Aspect Projections, the values h, k, a, and b have a very important relation: as sets, {h, k} = {a, b}. Stating this another way, if we look at the north-south scale h and at the east-west scale k, one of them will be the maximum scale a, and the other will the minimum scale b. Note that at a point, a and b do not have to be different. If the maximum scale a at a point equals the minimum scale b at that point, then all scales in every direction at that point are the same: a = h = k = b.
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Map projections with unnecessarily large distortion may confuse and mislead readers.
Based on the two maps of Ohio in Figure 3, a reader will wonder if Ohio is wider than it is tall. The distortion in the map on the left exceeds 20%. In fact, Ohio extends about
210 miles from east to west, and its north/south extension is 230 miles, making Ohio about 10% taller than it is wide.
Figure 3. Maps of Ohio using different projections (cylindrical on left, conic on right).
Different projections of the same region can appear quite dissimilar, and perhaps more importantly, the relative lengths of inter-point distances on one projection may vary greatly with relative lengths on another projection. See the size and shape of the triangles connecting the same three locations on six different projections of South America in
Figure 4. The relative lengths of the three edges representing the three inter-point distances of the three locations are totally disproportional.
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Figure 4. Maps of South America using different projections.
2.2 Scales Calculation
Normal Aspect Projections send meridian curves to straight lines. When the projected surface is collocated in the Normal Aspect position, each meridian curve maps onto a line that lies in the same plane as the meridian curve and that is tangent to the meridian curve at a single point on the curve
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Figure 5. Normal aspect projections' scales.
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One important class of Normal Aspect Projections that minimize distortion are the
Equidistant Normal Aspect Projections.
Figure 6. Equidistant normal aspect projections' scales.
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Note that for normal aspect projections, the maximum and minimum scales at any point are in the cardinal directions (north/south and east/west). Moreover, for the Equidistant
Normal Aspect Projections, the scale along meridian curves is 1 and 1 is the minimum scale. The maximum scale at every point is always in the east/west direction, and is given by k.
Oblique projections take on the same collection of scales as their normal aspect counterparts with respect to a rotated graticule; and the principal directions of the oblique projection are along the curve lines of the rotated graticule.
2.3 Normal Aspect Projections
Normal aspect projections have always been widespread and predominant because they are easy to compute. Their longitude/latitude pattern is very familiar and easily recognizable. Since we can assume that our minimum scale is 1, we only need to consider the tangent position, which simplifies calculations, thus reducing cartographers’ workload. Traditionally, projections have also been classified as equidistant, equal-area, or conformal according to properties that they preserve. If as before, we let h represent the scale in the north-south directions and k represent the scale in the east-west directions, we overall have 9 different basic normal aspect projections as Table 1.
Table 1. Classification of normal aspect tangent projections.
Normal Aspect Azimuthal Conic Cylindrical Equidistant h = 1 h = 1 h = 1 Equal-area hk = 1 hk = 1 hk = 1 Conformal h = k h = k h = k 10
Since my research will only focus on tangent projections, all the “normal aspect tangent projections” have been simply called “normal aspect projections” in this paper.
2.3.1 Normal Aspect Azimuthal Projections
Normal aspect azimuthal projections align the datum surface’s polar axis with the plane’s normal at the tangent point on the pole, producing a signature graticule pattern of concentric circles and equally spaced straight lines or line segments radiating from the common center. Thus, it merely suits for circular polar regions such as the Antarctic
Continent.
Among all the normal aspect azimuthal projections, the best azimuthal projection for minimizing distortion is the equidistant azimuthal projection.
In fact, for circular regions on the sphere, Milnor (1969) has proved that the projection that minimizes distortion is the azimuthal equidistant projection tangent at the center of the circular region [1].
2.3.2 Normal Aspect Cylindrical Projections
A normal aspect cylindrical projection produces a signature graticule pattern of equally spaced vertical lines and not necessarily equally spaced horizontal lines.
Although it preserves the orthogonality between longitude and latitude, it only has low distortion for equatorial regions due to the rapidly increasing scale k as one moves away from the equator. A normal aspect cylindrical projection, defined between -ϕmax and ϕmax, will have maximum scale at ϕmax when k = 1/cosϕmax (provided, of course, that h ≤
1/cosϕmax). Since k depends only on the latitude, achieving the upper bound of maximum scale at the greatest latitude, both the equidistant and the Mercator are minimal distortion 11 projections. For the equidistant, h = 1; and for the Mercator, h = k = 1/cosϕ; and in either case, the minimum scale will be 1 and the maximum scale will be 1/cosϕmax.
If we choose the equidistant projection, we can preserve the distance along meridians. If we choose Mercator, we can reduce distortion within small subregions away from the equator. Because it is easier to implement and analyze, we have chosen to work with the equidistant cylindrical projection as our representative of minimal distortion cylindrical projections.
2.3.3 Normal Aspect Conic Projection
We can treat normal aspect conic projection as a transition from azimuthal to cylindrical projection, and the flattened cone exhibits a signature graticule pattern of concentric circular arcs all of the same angular extent and equally separated straight lines or line segments radiating from the common center.
Compared to azimuthal and cylindrical projections, the best conic projection for minimizing distortion is much more complicated.
The normal aspect conic projection tangent at latitude ϕ0 that has minimum distortion is made up of two different projections glued together at the tangent parallel: the minimum distortion projection is the equidistant conic projection for latitudes below ϕ0, and is not the equidistant conic projection for latitudes greater than ϕ0. The conformal conic projection has much less distortion than the equidistant conic projection for latitudes greater than ϕ0.
Below ϕ0, to minimize the growth of k as we move away from the tangent circle, we need to keep h as small as possible, which in our case is h = 1.
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Above ϕ0, there is a tradeoff in the growth of h and k. To slow down the growth of k, we can let h take on some values bigger than 1, although those values of h should not get bigger than the maximum achieved by k itself, or h will become the maximum scale. To keep h growing at exactly the same rate as k, we can try conformality, since h and k both start at 1 at the parallel ϕ0.
To simplify the problem, however, we have chosen to work with the equidistant conic projection as our near-minimizing object.
2.4 Oblique Projections
A common reason for tilting a projection surface on the sphere is to move a larger, important region to be mapped to the places of lesser distortion. In contrast to normal aspect projections, oblique aspect projections align some pair of non-polar antipodal points on the datum surface with the projection surface’s axis (in the case of conic or cylindrical projections) or with the projection surface’s normal (in the case of azimuthal projections). If the azimuthal projection surface is a plane, the plane will be tangent at one of the antipodal points.
For any region on the sphere, we define the region’s best-fitting circle to be the circle that minimizes the region’s points’ maximum distance from that circle. So, for azimuthal projection, the best-fitting circle is the smallest containing circular spherical cap; for cylindrical projections, it is the best-fitting great circle passing through the region; and for conic projections, it is the best-fitting lesser circle passing through the region. We now build tables and draw graphs to show amounts of distortion in equidistant projections.
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From Table 2, we can see that when the radius of circular region gets larger, the distortion change is not a linear function of the radius.
From the Table 2, when we increase distortion value slightly from 1% to 2%, the corresponding radius of circular region increases 636.570 KM, while it only increases
483.380 KM when distortion reach 3% from 2%. The nonlinearity indicted in Table 2 is supported by Figure 7.
Compared to azimuthal equidistant projection, azimuthal stereographic projection’s performance is obviously worse with a sharp increasing curve.
Table 2. Distortion for the equidistant azimuthal projection of a circular area.
Radius of Distortion Distortion Corresponding
circular region (Unit: %) radius (KM)
10 km 0.00004106 0.01% 156
20 km 0.00016425 0.05% 349
50 km 0.00102654 0.1% 493
100 km 0.00410626 0.2% 697
200 km 0.01642644 0.5% 1102
300 km 0.03696480 1% 1555
500 km 0.10272726 2% 2192
1000 km 0.41179710 3% 2675
1500 km 0.92989106 5% 3430
2000 km 1.66153703 10% 4772
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Figure 7. Distortion = ((x/R)/sin(x/R)-1)*100.
Figure 8. Distortion = ([2tan(x/2R)/sin(x/R)]-1)*100.
In Table 3, for a given width of band region around a great circle, such as 20 KM, normal aspect cylindrical projection provides us 0.000492% distortion, larger than normal aspect
15 azimuthal projection’s 0.000164% in Table 2. The distortion for a 3000 KM width band region, the distortion is 2.837%. If we want to get 10% distortion, we have to increase corresponding band width to astonishing 5475.234 KM. Thus, cylindrical projection cannot fit wide regions well.
Table 3. Distortion for cylindrical projections of band around a great circle.
Half width of Distortion Distortion Corresponding half
band region (Unit: %) of band width (Unit: KM)
10 km 0.00012318 0.01% 90.096
20 km 0.00049274 0.05% 201.427
50 km 0.00307968 0.1% 284.801
100 km 0.01231968 0.2% 402.602
200 km 0.04929390 0.5% 635.777
300 km 0.11096824 1% 897.265
500 km 0.30875266 2% 1263.717
750 km 0.69693451 3% 1541.432
1000 km 1.24461464 5% 1974.020
1500 km 2.83713614 10% 2737.617
We also observed the nonlinear increasing rate of distortion, supported by Figure 9.
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Distortion = (1/cos(x/R) -1)*100
Figure 9. Distortion of a given width band region (Unit of x: KM).
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Table 4. Distortion for conic projections of band around a lesser circle.
Distortion for a tangent cone with cone angle: (Unit: %)
Width of band 10◦ 45◦ 80◦ region 10 km 0.0000307932 0.0000307825 0.0000307051
20 km 0.0001231615 0.0001230556 0.0001224597
50 km 0.0007695509 0.0007678996 0.0007587319
100 km 0.0030768452 0.0030636945 0.0029921924
200 km 0.0122970076 0.0121927417 0.0116482562
300 km 0.0276464042 0.0272975907 0.0255451909
500 km 0.0766863301 0.0750988353 0.0676842826
750 km 0.1722881493 0.1670388223 0.1444936711
1000 km 0.3059348383 0.2937316680 0.2452066214
1500 km 0.6874235210 0.6476937799 0.5100190283
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Table 5. Distortion for conic projections of band around a lesser circle.
Band width for a tangent cone with cone angle: (Unit: KM)
Distortion 20◦ 40◦ 60◦ 80◦
0.01% 180.501 180.903 181.657 184.941
0.05% 404.396 406.400 410.138 426.137
0.1% 572.684 576.679 584.106 615.423
0.2% 811.355 819.310 834.018 894.619
0.5% 1286.864 1306.548 1342.540 1483.458
1% 1824.964 1863.825 1933.912 2191.681
2% 2587.656 2663.774 2798.188 3249.425
3% 3172.325 3284.484 3479.162 4089.225
5% 4094.477 4275.655 4581.318 5445.243 10% 5756.357 6095.695 6638.426 7939.160
Figure 10. Distortion of a given width band region (Unit of x: KM).
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Figure 11. Distortion of a given width band region (Unit of x: KM).
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2.5 Comparison of Normal and Oblique Aspect Projections
For every oblique projection of the sphere, there is a point P0 on the sphere (at latitude 0 and at longitude 0) that corresponds to the North/South Pole at (0, 0, ±R) for Normal
Aspect Projections.
Table 6. Comparison of normal aspect projection VS. oblique aspect.
Normal Aspect Projection Oblique Aspect Projection Orientation point North/South Pole: P0 = (R cos0 cos0, (OP) NP = (0, 0, ±R) R cos0 sin0, R sin0 ) Projection surface Polar axis Line passing through P0 axis and -P0 Geodesics from OP Μ=Meridian curves Γ=Great (semi-)circles starting at P0 Points at same Π=Parallels of latitude Λ=Lesser circles with centers distance to OP on line (P0, -P0) Curves in principal Graticule = Μ Π Γ Λ directions
Figure 12. Normal VS. Oblique aspect azimuthal projected map.
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Figure 13. Normal VS. Oblique aspect cylindrical projected map.
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Figure 14. Normal VS. Oblique aspect conic projected map.
Figure 12, Figure 13, and Figure 14 (P0 = (16° E, 48° N)) provide us a direct and intuitive way to understand how the meridians and parallels look on different projections. The upper layer light mint green graticule-like lines represent the projected great circles and lesser circles associated with the orientation point P0, following the same pattern of normal aspect projection’s graticule on the left. The bottom layer black graticule is the actual set of meridians and parallels under oblique projections, not perpendicular to each other when they are not aligned with the principal directions. Among three different oblique projections above, the distortion of the oblique cylindrical projection increases fastest as one moves at the same rate away from the tangent point or circle.
2.6 Normal and Oblique Aspect Projection’s Equivalent Properties of Distortion
On a sphere, we classify lesser circles by their size. Any lesser circle will have the same size as one parallel of latitude at some latitude ϕ > 0 (in the Northern Hemisphere). We label such circles Cϕ.
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If 0 < ϕ* < ϕ are two latitudes, and if Cϕ* and Cϕ lie in parallel planes and in the same hemisphere, then the region between Cϕ* and Cϕ is isomorphic to the region between the parallels of latitude at ϕ* and ϕ. Any normal aspect conic projection defined on the region between parallels at ϕ* and ϕ has a corresponding oblique aspect projection defined equivalently on the region between Cϕ* and Cϕ.
Thus, the property of distortion for a region that lies between two lesser circles will not change at all no matter which kind of projection we finally take, normal aspect or oblique aspect. The only decisive factor is the shape of this region, since we are always trying to find two closest lesser circles, same as parallel planes in 3D space, encompassing it.
Cϕ ϕ Cϕ * ϕ*
Figure 15. Projection's property of distortion.
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Chapter 3. Oblique Azimuthal Projection
3.1 Methodology
Figure 16. Oblique azimuthal projections.
As we know, the maximum scale in the equidistant azimuthal projection occurs at points at the greatest distance from the tangent center on the sphere, all the points on any fixed circle centered on the tangent point must share the same max and min scales; and the outermost circle decides the globally largest maximum scale. So the question of finding the best fitting oblique azimuthal projection changes to finding the best tangent point which minimize the outermost position’s distance.
The simplest and most direct way is to test an increasingly dense set of candidate center covering radii to find the best tangent point. Although modern high-performance
25 computers allow us to do that, we want to figure out more ingenious and efficient algorithm to solve this problem.
For a given area, as I mentioned above, the largest scale is decided by the furthest boundary point, thus transforming the problem to finding a minimum circle or minimum spherical cap that encompasses the given area on the sphere.
Figure 17. Minimum fitting circle in 2D plane.
Take the point set in a 2D plane as an example to illustrate the circle expansion theory to find the minimal circle encompassing all the points.
As the Figure 17 illustrates, we first find the two points P1 and P2 of greatest separation, and draw a black circle with diameter [P1, P2], then check whether all the other points are 26 located inside the black circle or not. If yes, this is the minimal circle we are looking for.
Otherwise, we find the farthest point P3 to the circle center and expand the current black circle to be big enough, the red circle, to include this outer point P3 as well as P1 and P2.
Then we conduct the same validation check to this new larger red circle. But now, when we keep expanding the current red circle, we have 2 choices. One is the blue circle passing P1, P3, and P4. The other is the green circle passing P2, P3, and P4. However, we will only keep the smaller circle passing P4, that is the blue one. Conducting validation check, we find this blue circle satisfies our termination condition. If not, we will keep expanding it. After several iterations, we can finally get the circle satisfying our termination condition.
Below is the pseudo-code of this strategy.
Step1: Find the two points which are furthest apart and draw the circle having those two points as end points of the circle’s diameter
Step2: Check whether all the points are inside this circle (using determinant)
Step3: If yes, return this circle and stop
Step4: If not, take the current circle’s diameter as a fixed chord and expand the circle to encompass the point furthest from the center of this chord
Step5: Back to Step2
Rethinking this problem, we turn our attention to vector dot products, which provide us better solution.
If we use VN to represent the candidate directional vector of projection surface axis and use Vi to represent the directional vector of each boundary point in the region D.
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DPRes = { VN · Vi | Vi in the boundary of the region D }
Thus, we can get the range of the above dot products set. The minimum value of DPRes will occur at the points having the maximum scale for this projection surface axis VN.
After seeing the relationship between directional vectors and dot project, this problem becomes much easier. Finding the best projection axis means maximize the minimum value of DPRes (VN) among all the possible candidate directions VN.
Max({ Min( DPRes) (VN in all directions)}).
Figure 18. Dot products of directional vector and boundary point vector.
3.2 Experiments
3.2.1 Data Preprocessing
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Figure 19. Basic information of input data.
The countries’ boundary shape files are all downloaded from http://www.diva-gis.org, an open source website providing basic administrative areas, roads, inland areas, land cover, and population geographic dataset. Since our research merely needs boundary data, I download administrative areas shape file of several research countries.
All the preprocessing is conducted using ArcGIS 10.3. When we open ArcGIS 10.3, we can check the properties and basic information of our downloaded China dataset as
Figure 19. The geographic coordinate system and datum are all based on WGS_1984.
For convenience, I filter all the isolated islands and only retain the mainland of China for research. Since China’s boundary has too many curves, we must simplify it using the polyline simplification method, called Bend Simplify, which is able to preserve the shape and extension of given polyline after simplification.
Next step is to use Feature Vertices To Points providing in ArcGIS 10.3 to extract featured points from simplified polylines. 29
Figure 20. Extract feature points from the simplified boundary of China.
In Figure 20, the sampled hundreds of feature points preserve the shape and extension of
China’s sinuous boundary very well.
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ArcGIS 10.3 also provides a Geometry Calculator to generate the geographic coordinates for those sampled feature points, still picking WGS_1984 as new data’s coordinate system.
Our last step is to export those points’ geographic coordinates as a txt file, which can be done in ArcGIS 10.3 easily.
Finally, we conduct all the above operations step by step for each research country’s boundary data.
3.2.2 Main Programs
For all the implementation, I chose Python as my programming language considering its simplicity and usability even though performance is not as high as Java or C++. Because
I will use some open source Python packages only available in Python 2.x, such as mpl_toolkits.basemap, I set Python 2.7 as my research environment. During the whole process, I mainly used these following Python 2.7 packages: sys, math, pylab, matplotlib.pyplot, mpl_toolkits.basemap, and numpy.
Following is the pseudo-code of my program for finding the best normal vector of oblique azimuthal projection.
Step1: Generate candidate directional vectors in a neighborhood N with density D.
Step2: For each candidate vector, calculate n dot products of all the n boundary points, and pick the minimum of those n dot products to represent that candidate vector.
Step3: Find the vector having the largest representative minimum dot product.
Step4: If this vector improves the maximum dot product results significantly, refine the neighborhood N and increase the density D in Step1.
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Step5: If the representative minimum stabilizes (converges), terminate and print the result.
For the first iteration candidate vectors generation, the neighborhood N is the whole globe and the density D is every 5 degrees along the Longitude and Latitude, overall
2592 vectors. Since the Earth is a continuous surface and the dot product is a continuous function in both variables, the change of maximum dot product should be small when we move the candidate vector slightly. That is to say, from the second iteration the new generated candidate vectors taken with greater density D from a small neighborhood N around the best vector from last iteration will home in on a vector that maximizes its minimum dot product step by step. Finally, we will get a vector maximizing dot product under our pre-set tolerance.
In my research, we suppose the Earth is a sphere and set its radius to 6371.00 Kilometers.
All the Geographic and Cartesian coordinates of any point on the Earth is based on those
2 assumptions.
I also conduct some optimization to my program based on properties of the equidistant oblique azimuthal projection. It is not hard to find that those vectors, whose ending points located inside the research country boundary, must be better than those vectors, whose ending points located outside the boundary, because the former is able to increase the minimum dot product. This finding helps optimize our algorithm and improve the efficiency of my program a lot.
3.3 Results Analysis
I ran the program and got the results shown in Figure 21.
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Figure 21. Running time results of oblique azimuthal projection by calculating dot product.
The testing data is China_Bd.txt, among the 2592 candidate vectors of the first iteration, the vector, maximizing dot product minima of all the boundary points, Cartesian coordinates are (-1263.158, 4714.171, 4095.200) KM, and its Longitude/Latitude are
(105° E, 40° N). Also I print the range of dot product of each iteration for better observation.
From the 3 iterations in Figure 21, we can clearly find the increasing trend of lower bound of projected band from 5819.314 KM to 5863.513 KM. Also, the increasing rate slows down gradually, which indicates getting closer and closer to the final best vector we are looking for.
This directional vector obtained above defines our oblique azimuthal projection. In order to analyze the distortion, I calculate and print the maximum scale. Since we use the equidistant oblique azimuthal projection, h=1, we only need to take k into consideration based on the formula k = ( π / 2 - ϕ) / cosϕ.
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We find that under this directional vector, the maximum angle between boundary points and this vector is 23.03 degrees, and this point’s distortion is 2.74%. This is a very small distortion after we adopt the oblique projection instead of traditional normal aspect projection to mainland of China.
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Chapter 4. Oblique Cylindrical Projection
4.1 Methodology
Figure 22. Oblique cylindrical projection.
Considering oblique cylindrical projections expands and promotes the usage of traditional normal aspect cylindrical projection greatly, by allowing any great circle, not just the equator, to be where the cylinder and sphere are tangent. Since the cylinder is tangent with the Earth on a great circle, it means there is no distortion at all along the great circle in the tangent direction. We aim to minimize the distortion of oblique cylindrical projection by minimizing the distance on the sphere of the farthest boundary points to the candidate tangent great circle.
Inspired by the method we used to find the best oblique azimuthal projection, we formulate an equivalent problem involving a dot product with a candidate normal vector.
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Figure 23. Dot products of directional vector and boundary point vector.
If we use VN to represent the candidate directional vector of projection surface axis and use Vi to represent the directional vector of each boundary point in the region D.
DPRes = { VN · Vi | Vi in the boundary of the region D }
Thus, we can get the range of the above dot products set, and use that range to determine the maximum distortion for this projection surface axis VN.
Finding the best projection axis means finding the vector minimizing the range of DPRes and make it separate symmetrically on the two sides of tangent great circle among all the possible directions.
Min({ Max(DPRes) - Min( DPRes) (VN in all directions)}) U (Max(DPRes) +
Min(DPRes))/2 <= ε (ε very small).
As I mentioned above, the maximum distortion is decided by the points on the boundary furthest from the great circle or finding a narrowest band encompassing given boundary,
36 which minimizes the maximum perpendicular distance on the sphere. Until now, we transfer original problem to find a narrowest band taking advantage of properties of dot products. Narrowest band for vectors’ dot products places requirements on the maximum positive dot product and on the minimum negative dot product..
So I redesigned my algorithm to take two necessary and indispensable requirements into consideration together. The most important task is to minimize the projected band range.
Thus, for the termination condition, we firstly sort all the vectors according to their projected band ranges in nondecreasing order, then check the symmetrical property one by one. This algorithm guarantees the final vector we found minimize the range, and satisfy symmetrical requirement at the same time within given tolerance.
4.2 Experiments
4.2.1 Data Preprocessing
Based on our knowledge to the distortion properties of cylindrical projection, I pick
Chile, a well-known very narrow and south-north extended country, as my best case- study research object.
Following the same procedures I used for China, I filter out small isolated islands, simplify the boundary of Chile, extract feature points from the simplified boundary, generate geodetic coordinates of those points, and finally export the coordinates to a txt file.
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Figure 24. Extract feature points from the simplified boundary of Chile.
4.2.2 Main Programs
Following is the pseudo-code of my program for finding the best vector of oblique cylindrical projection.
Step1: Generate candidate directional vectors in a neighborhood N with density D.
Step2: For each candidate vector, calculate n dot products of all the n boundary points, and use the projected range (maximum subtracts minimum of those n dot products) to represent that candidate vector.
Step3: Sort the representative projected range of each candidate vector in a nondecreasing order.
Step4: Pick the vector having the minimum projected range under the constraint that the mid of it is as close to 0 as possible (less than preset tolerance).
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Step5: If this vector improves the minimum projected range significantly, refine N and
D, then return to Step1.
Step6: If the directional vector stabilizes (converges), terminate and print the result.
The most difficult part of this algorithm is the tolerance value for two termination conditions. If tolerances too small, we might miss some valid candidates and meet run time error occasionally, however, if tolerances too large, we are not be able to get the best result as close as possible to the truth value. So we have to balance the pros and cons and adjust our pre-set tolerances according to different countries. For Chile, after several tests and evaluations, I finally set range tolerance to 10 Kilometers and symmetrical tolerance to 0.5 Kilometers, which means the sum of maximum positive dot product and minimum negative one should be smaller than 0.5 Kilometers.
For comparison, I also design a program calculating distortion directly, getting close to the truth value step by step. Following is the pseudo-code.
Step1: Generate candidate directional vectors in a neighborhood N with density D.
Step2: For each candidate vector, calculate n distortions of all the n boundary points, and pick the maximum of those n distortions to represent that candidate vector.
Step3: Pick the vector having the smallest representative maximum distortion.
Step4: If this vector improves the maximum distortion results significantly, refine N and
D, then return to Step1.
Step5: If the representative maximum stabilizes (converges), terminate and print the result.
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This algorithm should be much more accurate because it evaluates distortions directly, instead of using dot products. So I use its result as a benchmark to evaluate my algorithm.
4.3 Results Analysis
Run the program and get the results as Figure 25.
Test data is Chile_Bd.txt, among the 2592 candidate vectors of the first iteration, the vector, satisfying two indispensable requirements, Cartesian coordinates are -5752.115, -
2682.255, 555.269) KM, and its Longitude/Latitude are (155° W, 5° N). Also I print the projected range, width, and middle of dot product of each iteration for better observation.
The range of dot products is (-326.220, 419.922) KM. After 2 more iterations, we get the final best directional vector, which decreases the range width from initially 746.142 KM
Figure 25. Running results of oblique cylindrical projection by calculating dot product.
40 to 648.077 KM and makes the middle of projected range gets closer to 0 from 46.851 KM to 17.620 KM.
This directional vector obtained above defines our oblique cylindrical projection. In order to analyze the distortion, I calculate and print it out. Since we use the equidistant oblique cylindrical projection, h=1, we only need to take k into consideration based on the formula k = 1 / cosϕ.
The final maximum distortion is merely 0.144%, since under oblique cylindrical projection, the farthest points to the tangent great circle is only 3 degrees, which realizes a very narrow tangent band encompassing Chile.
In order to prove my algorithm works well, I run the dynamic programming version oblique cylindrical projection program, and get the result as Figure 26.
Figure 26. Running results of oblique cylindrical projection by calculating distortion.
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Although, this program takes more iterations to find the best vector minimizing distortions of Chile, the vectors found in each iteration is very similar to my algorithm, and the final minimum range width is 645.355 KM, slightly better than 648.077 KM.
While the range middle is improved greatly from 17.620 KM to 0.083 KM, satisfying symmetrical requirement better. And the maximum angle of farthest boundary point to tangent great circle is now 2.9 degrees, instead of 3.074 degrees.
Most important is the efficiency of two algorithms, the running time of dynamic version averages 1081 milliseconds, while my algorithm only takes 316 milliseconds, reducing the time by 71%. Since our test data only has a few hundred boundary points, the performance of my algorithm will be more evident and non-negligible when the boundary point dataset becomes larger and larger.
In evaluating the accuracy, performance, and reliability of my algorithm, I see that it generates the equidistant oblique cylindrical projection within acceptable tolerance quite well.
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Chapter 5. Oblique Conic Projection
5.1 Methodology
Figure 27. Oblique conic projection.
Conic projections are the most prevalent and widespread projection in the world because most countries lie at the mid-latitude regions. However, using only the normal aspect conic projections, one cannot depict some extended counties very well. Countries such as India, Angola, and Poland are not located exactly at mid-latitudes, so normal aspect conic projections are not very suitable. Oblique conic projections, however, prove more suitable since we can rotate the axis of projection to make it fit the shape and extension of given regions more adequately.
Compared to oblique azimuthal projections and oblique cylindrical projections, oblique conic projections are more complex, because even after we find the best oblique directional vector for a conic projection, we still need to find the tangent lesser circle. 43
Both the cone axis and the tangent lesser circle together are needed to fully define our oblique conic projection.
Encouraged by the success achieved for oblique azimuthal and cylindrical projections, I still try to explore ways to solve this problem following the idea of dot product.
If we use VN to represent the directional vector of projection surface axis and use Vi to represent the directional vector of each boundary point in the region D.
DPRes = {VN ·Vi | Vi in the boundary of the region D}
Thus, we can get the range of the above dot products set. Finding the best projection axis is related to minimizing the range of DPRes.
Min({ Max(DPRes) - Min( DPRes) (VN in all directions)}).
Minimizing the distortion means finding a narrowest band encompassing the given region and the maximum geodesic sphere distance to the tangent lesser circle should be as small as possible. I started by constructing some simple examples to find trends and properties of conic projections. By considering the distortion directly for a given fixed latitude range, I found that the best tangent lesser circle always locates near but not at the middle of this range. Also, when this fixed width band move towards the pole, the best fitting tangent position of lesser circle will move more towards the pole within the band. But after testing several times, I did not find any simple way to describe the tangent circle position. Because I could not modify my dot product methods to work for cones, I decided to use distortion directly, as I did for the dynamic version of the oblique cylindrical projection determination even though that performance was limited.
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5.2 Experiments
5.2.1 Data Preprocessing
Considering the distortion property of conic projection, we look for a trapezoidal country in the mid-latitude region, a country like Italy. Because Italy extends in the northeast and southwest direction and the northeast region is narrower than the southwest, an oblique aspect conic projection should have less distortion than a normal aspect conic.
Following the earlier procedures, I extract feature points from the simplified boundary, and finally export the coordinates of them to txt file.
Figure 28. Extracted feature points from the simplified boundary of Italy.
5.2.2 Main Programs
Following is the pseudo-code of my program for finding the best fitting oblique conic projection. For each candidate directional vector, I test 1,000 possible tangent positions.
Step1: Generate candidate directional vectors in a neighborhood N with density D.
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Step2: For each candidate vector, calculate n distortions of all the n boundary points, and pick the maximum of those n distortions to represent that candidate vector.
Step3: Pick the vector having the smallest representative maximum distortion.
Step4: If this vector improves the maximum distortion results significantly, refine N and
D, then return to Step1.
Step5: If the representative maximum stabilizes (converges), terminate and print the result.
For Step1, we still first generate all the 2592 potential directional vectors in every direction and then generate new vectors based on the best vector found in the previous iteration. I pre-set the number of possible tangent positions to 1,000, that is to say the program will test 1,000 different tangent positions for every directional vector and only store one tangent position, which minimize the maximum scale, for that particular vector.
5.3 Results Analysis
The test data is Italy_200.txt, among the 2592 candidate directional vectors of the first iteration, the vector, minimizing the maximum distortion, Cartesian coordinates are
(3137.105, 2632.344, 4880.469) KM, and its Longitude/Latitude are (40° E, 50° N).
Also, I print out the maximum distortion of every iteration for analysis. After 3 more iterations, we get the final best directional vector, which decreases the maximum distortion slightly from 0.178% to 0.173%. And the final vector’s Longitude/Latitude are
(34.48° E, 48.96° N). Besides, I print out the new “Latitude” under oblique projection for analysis. From Figure 29, we can see this region is from 68.97 to 75.70 in degrees.
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Figure 29. Running results of oblique conic projection by calculating distortion.
This means the narrow oblique band encompassing Italy is only less than 6 degrees of latitude. And the best fitting lesser circle is tangent at 72.55 in degrees instead of exactly at the middle of 68.97 and 75.70, which would be at 72.34. This result corresponds with my findings that the best fitting lesser circle has the trend of moving towards the upper part of the band as the band moves toward polar regions.
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Chapter 6. Evaluation
6.1 Evaluation of Performance
I have illustrated the algorithms for finding the best fitting oblique equidistant azimuthal, cylindrical, and conic projections in the last 3 chapters in detail. In this chapter, I will focus on evaluating the accuracy and reliability of them.
Distortion is the most important and fundamental criteria when we evaluate the performance of a new projection algorithm, so I firstly calculate and list the maximum distortion of each projection algorithm for nine distinctive test countries as Table 7.
Table 7. Distortions under different equidistant oblique projections.
Country Oblique Oblique Oblique Equidistance Equidistance Equidistance Azimuthal Cylindrical Conic Projection (%) Projection (%) Projection (%) China 2.74278905074 4.80051194912 3.60384996417 Chile 1.87976881614 0.12857414269 0.0786919416382 Italy 0.157566348816 0.194300786999 0.17362502419 United Kingdom 0.173635485033 0.0768541275902 0.0759008895689 Vietnam 0.27418619913 0.128436358784 0.0752357444333 Finland 0.13737409391 0.0944176340741 0.0789622828387 Greenland 0.730827887061 0.533224014951 0.496709433776 Indonesia 2.96077177611 0.738467642663 0.672791647657 Sweden 0.257487717473 0.0751272891567 0.0609806818776
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For China, the oblique azimuthal projection provides the minimum distortion as low as
2.743%, even though I thought the oblique conic projection might be the best since most of China’s maps use normal aspect tangent or secant conic projections. This result is caused by the rounded shape of China’s boundary. Also, we could treat China as a trapezoidal, which explains why the oblique conic projection provides worse distortion compared to azimuthal projection but better than cylindrical projection.
Thanks to Chile’s long, narrow, and slightly curved boundary, oblique conic projection obviously dominates the other with the distortion as low as 0.079%, which is very close to 0, means no distortion. There is no doubt that for Chile the oblique azimuthal projection is the worst of the three with distortion up to 1.880%.
The oblique azimuthal projection and the oblique conic projection provide similar distortion results for Italy, because Italy includes several islands that make it more or less fill up a circular region. Even we use oblique cylindrical projection for Italy, the distortion is still relatively small. In fact, all the three algorithms work well and provide us acceptable distortion results, which is caused by the comparatively small national territorial area of Italy. If a country’s size is large or its boundary is long enough, the differences among those three algorithms could be much conspicuous. That is why we research on different oblique projection algorithms.
The United Kingdom is very similar to Italy. Its extent runs mostly from northwest to southeast. Due to the small national territorial area and extension of this country, oblique conic projection provides minimum distortion result as low as 0.076%, which is even
49 smaller than the minimum distortion of Chile. However, the distortion of oblique azimuthal projection and oblique cylindrical projection are both still acceptable.
From Table 7, we can easily pick the best fitting projection for a region. For those countries with small territorial areas, like Italy, United Kingdom, Finland, and Sweden, the difference among three oblique projection algorithms is not that big. However, for those countries with vast areas, like United States and China, examining oblique projections is very helpful and significantly better at guaranteeing high accuracy maps.
6.2 Evaluation of Effectiveness
Visualizing distortion of the projected region on a map is an effective and convincing way to validate the efficacy of our algorithms. By calculating the scale (k) for all the test countries under three different oblique projections, I used a heat map to visualize the scale value for any position directly. The darker the color, the smaller the scale. The resolution of the heat maps are all 0.5 cm, and the map scales are all 1: 10,000,000.
From four figures of each test country in Figure 30 to Figure 34, we can easily find that the separation and pattern of scale value are exactly consistent with what we expected.
All the heat maps of different test countries are consistent with the distortion results on
Table 7 and our knowledge to the patterns for those oblique projections.
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Figure 30. Heat maps of distortion in China under different projections 51
Figure 31. Heat maps of distortion in Chile under different projections.
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Figure 32. Heat maps of distortion in Italy under different projections. 53
Figure 33. Heat maps of distortion in Vietnam under different projections. 54
Figure 34. Heat maps of distortion in Greenland under different projections. 55
Chapter 7. Conclusions
7.1 Discussion on angle and area distortion
Although we defined distortion in terms of global maximum and minimum scales, angle distortion and area distortion are two additional concepts we can easily define in terms of local maximum and minimum scales. We can define and understand both most easily by examining the Tissot Indicatrix, an ellipse of the scales in all directions in the tangent plane at any single point.
-1
b1a 1+ Figure 35. Tissot indicatrix and map projections.
Angular changes from datum surface to projection surface are also necessarily very small because the Tissot Indicatrix is an ellipse with very little flattening.
Maximum directional change from datum surface to projection surface for the azimuthal equidistant projection occurs at points of greatest scale in a direction ±45° from the
56 principal directions at the point. If the maximum scale is x, then the direction that is 45° from the principal direction corresponding to the minimum scale on the sphere becomes arctan(x). If x =1.01, which corresponds to 1% distortion, then 45°= arctan(1) becomes
45.3°= arctan(1.01). Similarly, we have -45°= arctan(-1) becomes -45.3°= arctan(-1.01).
Thus, a pair of orthogonally crossing curves (in the directions +45° and -45° from the principal directions) will cross at angle 90.6°. Moreover, this difference of 0.6° = (90.6°
- 90°) is the maximum crossing angle change for any pair of crossing curves. For a 2% distortion, the angles ±45° would transform into the angles ±45.57° = arctan(±1.02), changing curves crossing at 90° to curves crossing at 91.14°, producing a crossing angle increase of 1.14°, a crossing angle increase of less than 1.3%.
Most of the states in the continental US can be mapped by the equidistant azimuthal projection with distortion of 0.1% and a few require a radius with 0.2% distortion. These distortions change a 45° direction to arctan(1.001) = 45.03° and arctan(1.002) = 45.06° respectively. The greatest crossing angle change possible is for an angle of 90°, and one
90° crossing angle changes to 90.06° for the smaller states and to 90.12° for the 3 largest states.
The product of the scales a and b in the principal directions at any point gives us the area scale: ab at that point. If b = 1 (the situation for all equidistant projections), and if the maximum scale a equals (1+), then the area scale will also be (1+). Thus a 1% distortion produces an area scale between 1.0 and 1.01, and a maximum area increase of
1%.
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7.2 Practical significance of oblique projections
We list the US states fitting within given radius spherical caps in Table 8 to give readers a direct intuition on how large these states are and how much we can achieve after applying oblique projections. The number in the parentheses following the region’s name is the maximum diameter of it with kilometer as unit. Most of the US states fit within a spherical cap with radius 500 KM and distortion 0.1%. Even for Alaska, we can limit the maximum distortion of such a broad area to merely 0.5%.
Table 8. US states fitting within a spherical cap. Distortion Region Some US states that fit within the spherical Radius cap (unit of actual spherical cap radius: km) 0.01% 150 km Rhode Island, Delaware, Connecticut Massachusetts, New Hampshire, Vermont, New Jersey 0.02% 220 km Maryland, South Carolina, West Virginia
0.05% 350 km 23 of the 48 Continental United States Hawaii (320) 0.1% 500 km Idaho, Nebraska, Tennessee, Florida, North Carolina, Arizona, Oregon, Idaho, Montana, New Mexico, Colorado, Oklahoma 0.2% 700 km Montana (506), Texas, California
0.5% 1100 Alaska (1030) km
We also test several countries following the same operations and list the results in Table
9. Oblique projection realizes an ideal minimal distortion even for those countries, whose radii are more than 2200 kilometer like continental USA, China, and Canada, as low as merely 2%. 58
Table 9. Countries fitting within a spherical cap. Distortion Maximum Some countries that fit within the spherical Radius cap (unit of actual spherical cap radius: km) 0.01% 150 km El Salvador (131)
0.02% 220 km Belize (152), French Guiana (212),
0.05% 350 km Suriname (256), Guatemala (267), Nicaragua (291), Panama (325), Honduras (334) 0.1% 500 km Ecuador (367), Guyana (414), Venezuela (484)
0.2% 700 km France (540), Spain (566), Cuba (572), Italy (617), Ukraine (665) 0.5% 1100 km Madagascar (776), Algeria (1031), Thailand (1058), Peru (1066) 1% 1550 km Mongolia (1208), Congo (1230), India (1407), Kazakhstan (1495) 2% 2200 km Mexico (1670), Australia (2092), Chile (2114),
More than More than Brazil (2300), Continental USA (2340), China (2518), 2% 2200 km Canada (2781), Russia (4025)
The fundamental and practical significance of oblique projections is simple: oblique projections have much less distortion than the projections we are used to seeing. Our work can promote work by other researchers in this field. This can be a revolutionary refocusing of map projection theory.
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Bibliography
[1] Milnor, J., 1969. A problem in cartography. The American Mathematical Monthly, 76(10), pp.1101-1112.
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