The Mathematics of Perspective: An Introduction to the Cross Ratio
David A. Thomas
Department of Mathematics
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In the Projective Transformations section of this paper, you learned that the cross ratio may be computed from any perspective. Another way of expressing this idea is, “The cross ratio is invariant under projective transformations.” This fact has far-reaching consequences in computer graphics, architecture and design, and scientific visualization. Figure 1 shows a perspective view of a canyon system on Mars (Image courtesy of NASA). Selected landmarks are labeled with the letters A, B, C, X, D, and E. Superimposed on this image is a set of lines used to compute the cross ratio determined by rays AC, AX, AD, and AE. For reasons that will become apparent later, this cross ratio is computed indirectly using the line segments associated with points P1, P2, P3, and P4.
Figure 1 View 1 of a Canyon System on Mars In Figure 2, use the mouse key to move the Drag Point so that line AP3 intersects point X. ² If the top of the scene is north, where should the observer be located in order to achieve this perspective? ² What cross ratio is obtained? ² If you relocate line XY, what happens to the cross ratio? ² If you had a map view of the same canyon system and recalculated the cross ratio based on measurements of that image, what value would you obtain? ² If you had a different perspective view of the same canyon system and recalculated the cross ratio based on measurements of that image, what value would you obtain? Figure 2 Finding a Cross Ratio Associated with View 1 [GSP File] Figure 3 shows a different view of the canyon system, with a white box superimposed on the region containing point X. Your objective is to deduce the location of point X using your knowledge of the cross ratio. Using the mouse key, reposition the red line in Figure 4 so that AX points in the correct direction. ² Where should the observer be located in order to achieve this perspective? ² How do you know when the red line is correctly positioned?
Figure 3 View 2 of the Same Canyon System Figure 4 Using the Cross Ratio to Position Line AX in View 2 [GSP File] The specific problem illustrated in Figures 1 – 4 is abstracted and generalized in Figure 5. Assume that the two views correspond to two perspectives of some geographic region on Mars. In both views points A, C, X, D, and E represent the same landmarks. In View 1, position point X so that the indicated cross ratio is approximately 0.90. Then adjust the red line in View 2 to the same value. ² What does the red line in View 2 tell you about the location of point X? ² How could you use the ideas demonstrated here to identify the actual location of point X in View 2? What additional information would you need? Figure 5 General Problem [GSP File] This page uses JavaSketchpad, a World-Wide-Web component of The Geometer's Sketchpad. Copyright © 1990-2001 by KCP Technologies, Inc. Licensed only for non-commercial use. |
