The Mathematics of Perspective: An Introduction to the Cross Ratio
David A. Thomas
Department of Mathematics
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Dividing Space:
Non-Collinear Points In the Dividing Space: Collinear Points section of this paper, you learned that 1. The manner in which a set of four collinear points (A, B, C, and D) divides a given line segment (AD) may be characterized using the cross ratio. 2. The cross ratio may be computed either directly, using the segments themselves, or indirectly, using their projections on other lines. 3. If you change the relative positions of the four points, they divide the segment differently and their cross ratio changes accordingly. We now consider the manner in which a set of rays divides space. Consider rays EA, EB, EC, and ED in Figure 1. Note that points A, B, C, and D are not collinear. In this case, a cross ratio is computed based on the angles formed. Definition For
any set of lines EA, EB, EC, and ED, the cross ratio R characterizes
the partition of R =
Figure 1 Cross Ration for Non-Collinear Points [GSP File] ² Reposition point E, moving line XY as necessary. Compare the cross ratio computed based on the angles formed with the cross ratio computed using the segments created by points G, F, H, and I. State a conjecture comparing the cross ratio determined using angles and the cross ratio determined using segments. ² Reposition points A, B, C, and D so that they coincide with points G, F, H, and I, respectively. Then move point E. What do you notice?
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