The Mathematics of Perspective: An Introduction to the Cross Ratio
David A. Thomas
Department of Mathematics
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Dividing Space:
Collinear Points Artists and architects of the Renaissance succeeded in developing drafting techniques that correctly and realistically represented 3-dimensional objects on 2-dimensional surfaces. As seen in the History section of this article, drawings with these qualifies systematically resize line segments and angles to create the illusion of depth. For instance, in Figure 1, the edges and angles of the cube, which we understand are equal in 3-dimensional space, are resized to create the illusion of depth. Intuitively, it seems reasonable to search for a simple ratio expressing the relationship between segments HG and EF, GC and FB, and so on. Figure 1 Creating the Illusion of Depth [GSP File] ² Move the drag point along the horizon line and note how the segment lengths and their ratios change. Does any segment or ratio of segments remain constant for a fixed set of Vanishing Points? ² Move the drag point along the horizon line and note how the angles and their ratios change. Does any angle or ratio of angles remain constant for a fixed set of Vanishing Points? ² Move the Vanishing Points along the horizon line. Which measurements or ratios remain fixed? ² Move point E vertically. Which measurements or ratios remain fixed? In spite of the appeal of the idea that some ratio of segments of angles must be invariant under these perspective transformations, no such ratio exists. But surprisingly, this idea is only one step short of the truth. The sought for relationship is not a simple ratio but a ratio of ratios called the cross ratio. Definition For any four collinear points A, B, C, and D (See Figure 2), the cross ratio characterizes the partition of segment AD by points B and C, where R =
Figure 2 Cross Ration Computed in Terms of Segments [GSP File] |
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² Using the pointer, reposition points A, B, C, and D. What happens to the cross ratio computed in terms of segments BA, BC, DA, and DC? ² How does the cross ratio computed in terms of segments BA, BC, DA, and DC compare to the cross ratio computed in terms of the segments joining points E, F, G, and H? In terms of the segments joining points I, J, K, and L? ² Leaving points P, Q, A, B, C, and D fixed, reposition the segment containing points E, F, G, and H. What happens to the cross ratios computed in terms of those segments? ² Leaving points P, Q, A, B, C, and D fixed, reposition the segment containing points I, J, K, and L. What happens to the cross ratios computed in terms of those segments? ² Leaving points A, B, C, and D fixed, reposition points P and Q. Do the cross ratios computed in this manner depend on the positions of points P and Q? ² State a conjecture about the cross ratio based on your observations. |
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