CR6

The Mathematics of Perspective: An Introduction to the Cross Ratio

David A. Thomas

Department of Mathematics

University of Idaho

Martian Canyon System: In the Cross Hairs

In order to locate point X in View 2, we must know its direction from two reference points. In Figure 1, both point A and point B serve as reference points. Using the procedures presented in Martian Canyon System: Two Views, we may find a cross ratio for both the black and white line sets.

Figure 1 Two Reference Points [GSP File]

In Figure 2, each red line is positioned so that lines AX and BX, respectively, lead to the same cross ratios obtained in View 1. In effect, you locate point X with a “cross hair.”

Figure 2 Locating Point X [GSP File]

There is an aspect to this analysis that introduces a significant source of error: While the all of the lines are assumed to lie in the same plane, the landscape features marked A, B, C, X, D, and E do not.

² What effect might this assumption have on the accuracy of the result?

A low-tech version of this procedure called the paper strip technique has been used by cartographers (map makers) for decades,. Cartographers gather data from a variety of sources, including surveys, photographs, and satellite images. The cartographer’s job is to create unbiased map views that accurately identify significant landmarks (both natural and man-made), longtitude and lattitude, and other information (e.g., elevation, land use, population statistics, and so on). The paper strip technique is used to remove the bias inherent in the perspective views of photographic and satellite data sources. The steps imlementing this technique as applied to the Martian canyon system are as follows:

Step 1	Draw a set of lines from point A to points C, X, D, and E in View 1 (See Figure 3). Then lay a strip of paper across these lines and mark their intersection points.
Step 2	Draw lines AC, AD, and AE in View 2 (See Figure 4). Then position the paper strip from Step 1 so that the marks line up as shown. The unused determines the position of line AX.
Step 3	Repeat Step 1, drawing the set of lines from point B and using a second paper strip.
Step 4	Repeat Step 2, using the second paper strip to determine the position of line BX.
Step 5	Point X is located at the intersection of lines AX and BX.

² Why does the paper strip technique work?

² What is its mathematical basis?

Figure 3 Step 1

Figure 4 Step 2

Print out the images in Figures 5 and 6, then use them to practice the paper strip technique.

Figure 5 View 1 Practice Image

Figure 6 View 2 Practice Image

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