DISTORTION ON THE INTERRUPTED MODIFIED COLLIGNON PROJECTION Keith C. Clarke Department of Geologyand Geography Hunter College-CUNY 695 Park Avenue New York, NY 10021,USA and Karen A. Mulcahy Earth and Environmental Science Program Graduate School and University Center-CUNY 33 West 42nd St. New York, NY 10036, USA ABSTRACT The Interrupted Modified Collignon or "Clarke's Butterfly" projection is a near-equal area projection of the earth as a sphere onto eight equilateral triangles, arranged as octants. This projection was presented to solve a problem of mapping into an initial set of equilateral triangles that could then be divided recursively into triangular quadtrees based on Dutton's Quaternary Triangular Mesh. This projection was used as a test case for experiments in designing a method for the visualization of map projection distortion. We present a new method for portraying the distortion, in which the projection parameters of local shape, area, and direction are shown as several experimental combinations of red, green and blue color intensity representations. Some of these combinations show distinct promise as visualization methods for map distortion portrayal, and assist greatly in the understanding of the Interrupted Modified Collignon projection's spatial properties. Extension of the method by its incorporation into map projection software is discussed, so that map users will have a simple and effective means to visualize both global maps, and their inherent inaccuracies. INTRODUCTION In an earlier paper (Lugo and Clarke, 1995), a projection was described that maps the surface of the earth, assumed spherical, onto eight equilateral triangles, and arranged them as octants into a butterfly pattern (Figure I). The motive for the construction of the projection was the development of a global relief data structure based on triangular quad trees and two dimensional run length encoding, using Dutton's Quaternary Triangular Mesh system (Dutton, 1984). Advantages of the projection for cartography are the near equal area properties of the projection, plus the fact that the butterfly can be folded and pasted into a three dimensional octahedron, and even "assembled" visually from the layout of the octants. The projection built on a family of projections presented by Cahill in 1932. The major difference is that the current version has parallel, near-equally spaced straight meridians and straight converging parallels, and that the Collignon triangular projection, 1. --
, which is equal area, has been modified so that each octant is an equilateral triangle. In addition, the arrangement into a butterfly pattern and numbering of the octants differ slightly. This has the advantage of forming a foldable three dimensional figure that can be "sliced" at any parallel, or indeed any meridian with its opposite, to depict the earth's interior or atmosphere. Slices along parallels generate square maps getting smaller toward the poles, while slices along a meridian reveal a diamond shape. Figure 1: The Interrupted Modified Collignon (Clarke's Butterfly) Projection. @Copyright Keith C. Clarke, 1994 The formulae for the spherical version of the projection are as follows, The meanings of the variables is as standard, and the reader is referred to Snyder (1987). Each octant is processed separately, and is mapped into its butterfly position with the affine transformations listed in Table I. The arrangement of octants is shown in figure 2. x = 2RS (I...- 11.0)Jl J[i - sin</> y = cos(30) xj[ixrs(jl-sin</» Invert y for southern hemisphere. Table 1: Affine Transformations by Octant Octant rotation x translate y translate central meridian I 60 -S sin 30 S cos 30-135 2 0 0 0-45 3 300 2 S sin 30 0 45 4 120 0 2 S cos 30 135 5 60 -S sin 30 - S cos 30 135 6 120 0 0-135 7 180 2 sin 30 0-45 8 240 3 S sin 30 - S cos 30 45 C. L 1\11 i,!
Figure 2: Transformation from a) square-based graticule. to b) triangle-based format. to c) Butterfly Projection. a) (180-9090 0 (90.90) I 090) 1 2 3 4 (00) 6 7 I I (180.-90) (-90.-90) c) S b) The butterfly projection is a compromise, that is, it is neither conformal nor equivalent. In addition, being interrupted, a complex pattern of directional distortion is evident. As such, the projection is a good test case for exploring visualization methods of projections distortion. This paper builds upon the discussion of the projection by presenting a new visualization method for cartographic representation of the inherent distortion during the map projection transformation. The method uses simple mappings of the projection transformation scale and directional distortion onto an image, with the red, green and blue image intensities scaled proportionally to the distortion dimensions. The paper first introduces comparable methods, then develops the new method, reports upon an implementation for the Equatorial Mercator and Sinusoidal projections, and finally discusses the choice of color representation in the context of the more complex visualization of the error in the Interrupted Modified Collignon.. Distortion mapping for the map projection transformation has a long history. Robinson et al. (1995) report only three methods of visualization. These are (I) drawing isolines of the values used in Tissot's indicatrix, (2) drawing Tissot's indicatrix at points on the map and (3) showing the results of projecting a familiar figure such as the outline of the human head. This reflects the seminal work of Tissot (1881) in generating the first scientific measure of map projection distortion. In 1859 and 1881, Tissot published a method of depicting the distortion incurred during the map projectiontransformation,introducingthe graphicaldevice of the indica-. trix. The indicatrix, capable of placement at any point on a map, is simply a circle of very small size on the globe that is projected using the same method of projection as the map, and then rendered on the resultant map. Virtually all examples, such as those in Snyder and Voxland (1989), show the indicatrix on a blank graticule, and evenly spaced at the intersections of parallels and meridians. An example is shown in figure 3. The indicatrix in general can be thought of as a mapping of the circle on the globe onto an ellipse on the map. If the original circle is thought of as having a "north-south" and an "east-west" vector, centered at the point where the indicatrix is to be drawn, then on the map a new ellipse will be formed, with a size, shape and orientation determined by the projection. The ellipse size, divided by the size of the original circle at scale, is equivalent to the area distortion. This is given by the product of the scale factors along the axes of the iellipse. If these do not multiply to unity, the map is not equivalent. C L (::. ". \.c '.. --- -- - ---
Figure 3: Tissot's Indicatrixes drawn at the intersections of the graticule on the Lambert Conformal Conic Projection. Source: Snyder and Voxland. 1989.. Tissot emphasized the amount rather than the direction of the angular distortion. Tissot computed the value (J), the maximum angular distortion between any two orthogonal lines on the map and the same two orthogonals on the globe (Figure 4). When the maximum angular distortion is zero at all places on the map--such as on a cylindrical projection-the projection is conformal. Snyder (1987, p. 21) stated that "orientation is of much less interest than the size of the deformation." Nevertheless, the actual angle of the direction distortion based on the final map's orthogonal axes, is called the convergence or grid declination, and is of great practical interest to map users. Nevertheless, the convergence is very important to the visualization of distortion, as also are the scale and areal distortion. In the following discussion, the properties thought worthy of visual presentation were (I) the east-west extent of the indicatrix in the map coordinate space, (2) the north-south equivalent and (3) the convergence. In each case the values were not absolute, as Tissot used, but relative. The area distortion becomes a product of the first two values. Figure 4: Derivation of (J) from Tissot's indicatrix. Thin lines are the graticule. On the globe, there exist two pairs ofiines that each meet at right angles (grey and black) such that the same lines on the map show the maximum angular difference between the line that still meets at right angles (grey) and the other pair. The angle between these two orthogonal lines is the maximum angular distortion.
COLORDEPICTIONOF DISTORTION A computer program was written by the first author that (I) rendered a world outline into a given projection, (2) iterated throughout the relevant section of the globe at IS minute intervals, computing the east-west map distance of a slight displacement along the parallel centered on the point in question, the north-south map distance with the same displacement along a meridian, and the angle of convergence. These values were scaled to cover the range of one byte in each case, and the map coordinates plus the three color values were written to a file. (3) An additional computer program interpolated the point data to a grid, using the algorithm in Clarke (1995), and finally (4) the world outline and distortion maps were interwoven to produce a distortion image map. Convergence (Blue) GLOBE MAP Figure 5: Values computed for the visualization method. East- West Distortion (Red) As a test of the method, the Equatorial Mercator and Sinusoidal projections were processed, and the results shown in figure 6. """ w,.. 'ii?;o.. Figure 6: Rendering of distortion maps for the equatorial Mercator (upper) and Sinusoidal (lower) map projections. Red represents east-west distortion. green north-south. and blue the convergence angle. The Mercator Is conformal. so there Is no blue variation. and the Sinusoldalls equivalent. with no red or green variation. --- - ---
With the computer program functioning, the Clarke's Butterfly projection was also processed, resulting in the distribution shown in figure 7. The originals of these diagrams are in color, and cannot be depicted in a monochrome proceedings, but are available by searching for Map Projections on the Hunter College Home Page on the World Wide Web athttp://everest.hunter.cuny.edu:www. Figure 7: Distortion map of projection distortion for the Clarke's Butterfly projection. 1\vo alternate colorations are used. On the left. the same value as figure 6 are used. On the right. the colors are rotated (red is convergence. green is e-w and blue is nos distortion. As is also partly shown in figure 7, a set of six different versions of this map was produced, and can be viewed on the World Wide Web, reflecting each possible color combination of the three additive primary colors. Of these, by far the most preferable seems to be the simplest, that is the two spatial dimensions (scale, and area) are shown as red and green, and the convergence is shown as blue. The visual nature of blue, being associated as a cool color and with the oceans perceptually, and as a color with little contrast depth visually, made this combination most suitable. CONCLUSION In conclusion, we present a new method for the visualization of map projection distortion, based on mapping properties of Tissot's indicatrix and other distortion measures onto a color image depicting three variables simultaneously. The method has been used to depict distortion on two commonly used map projections, and also to help visualize and explain distortion on the Modified Interrupted Collignon projection, that varies in all three dimensions of distortion. The simplest color coding, red for east-west scaling, green for northsouth, and blue for convergence, seems to give the best visual results. Further research will first extend the method to other map projections. This will most likely take the form of the method's incorporation into the map projection educational package Mercator, under development as a student project at Hunter College. Similarly, many aspects of color use remain to be.explored. First, the values shown in color can be extended to include the scale factors, area distortion, and the angle of maximum angular deformation. All of these are computed from the Tissot's indicatrix. Similarly, much work C. 1.-.1\,'-.. Y( \;;... -- -- -- - --
remains in deciding which color representations, scaling and form of representation (RGB vs. HSI) are perceptually most suitable for depicting distortion. Finally, a color inset map showing projection distortion could be simply and effectively integrated into many computer and paper map products, especially at small scales, much like the reliability diagrams on topographic maps. Since map projection distortion is usually the major source of misunderstanding and error in map use and GIS settings, a visual cue to projection error could go a long way toward the better understanding and consequent minimization of cartographic error due to map projection distortion. REFERENCES Cahill; B. 1. S. (1932) "The Butterfly Map Projection," Nature, No. 3295, Vol. 130, pp 973-974. Clarke, K. C. (1995) Analytical and Computer Cartography, 2 ed., Prentice Hall: Englewood Cliffs, NJ. Dutton, G. (1984). "Geodesic Modelling of Planetary Relief." Cartographica, v21, Numbers. 2 & 3, pp. 188-207. Lugo, J. A. and Clarke, K. C. (1995). "Implementation of a Triangulated Quadtree Sequencing for a Global Relief Data Structure." Proceedings. Autocartol2, ACSM/ASPRS Annual Convention and Exposition Technical Papers, Volume 4, pp. 147-156. Charlotte, North Carolina, February 27-March 2,1995. Robinson, A. H. et al. (1995) Elements of Cartography, 6 ed. New York, J. Wiley. Snyder, J. P. (1987) Map Projections-A Working Manual. United States Geological Survey Professional Paper 1395. U. S. Government Printing Office, Washington D.C. Snyder, J. P. and Voxland, P. M. (1989) An Album of Map Projections. United States Geological Survey Professional Paper 1453. U. S. Government Printing Office, Washington D.C. Tissot, N. A. (1881) Memoire sur la representation des surfaces et les projections des cartes geographiques, Paris, Gauthier Villars. 7 - -- --