CR3

The Mathematics of Perspective: An Introduction to the Cross Ratio

David A. Thomas

Department of Mathematics

University of Idaho

Projective Transformations

In the Dividing Space: Collinear Points and Dividing Space: Non-Collinear Points sections of this paper, you learned that

² The division of space by points on a line or by angles may be characterized using the cross ratio.

² The cross ratio may be computed using collinear line segments or concurrent angles, depending on the circumstances.

² Applied correctly, both methods produce the same result.

In the examples considered so far, the planar figures are shown in what is sometimes called a map view, as if you were looking down on the figure from directly overhead. This is the conventional manner to represent figures in plane geometry, so it is from this perspective that we normally measure segments and angles.

We now turn our attention to perspective views of planar figures and ask the question, “If we measure the segments and angles in a perspective view of a figure and use those measurements to compute a cross ratio, will the result be the same as or different than that obtained using measurements based on a map view of the same figure?” This situation is illustrated in Figure 1.

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Figure 1 Perspective and Map Views of the Same Figure [GSP File]

² Using the pointer, drag around the Vanishing Point and point P along the horizon line. How does the perspective view of the tile floor change?

² How does your position as an observer appear to change from one projection to the next?

Using the segment-method, the cross ratio is computed based on the manner in which the tiles divide the heavy red line drawn in both the Map View and the Perspective View.

² What do you notice about these two cross ratios?

² State a conjecture about the cross ratio as computed from perspectives other than map views.

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