cross ratio - 4 CROSS RATIO 4.1 Euclidean Cross Ratio Very...

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53 4. CROSS RATIO §4.1 Euclidean Cross Ratio Very few quantitative properties of an object remain fixed when it’s drawn in perspective. For example the following is a perspective drawing of a railway track. The picture is very much smaller than the original, so lengths and areas are not preserved. The sleepers on the actual track are at right angles to the lines yet in the perspective drawing they’re not, so angles are not preserved. Ratios of lengths along a line are not preserved because the sleepers are equally spaced while they get closer and closer in the perspective drawing. But ratios of ratios along any line are preserved. Definition: Suppose A, B, C, D are four collinear points. Choose a direction along the line as the positive direction (so that distances measured in the opposite direction are treated as negative). The cross-ratio of these four points, in the given order, is defined to be: (A, B; C, D) = AC BC AD BD where AC, BC etc denote the signed distances between the respective points. (Clearly the cross ratio is independent of the chosen direction, but does depend on the order in which the points are taken.) Example 1: 3 1 2 A B C D (A, B; C, D) = AC BC AD BD = 4 1 6 3 = 2. As mentioned earlier, the cross ratio depends on the order of the points. In the above example (C, A; D, B) = CD AD CB AB = 2 6 - 1 3 = - 1. The fact (which we have yet to prove) that cross ratios are preserved when things are drawn in perspective means that if four collinear points are projected from any point onto another line, the image points have the same cross ratio as the original ones.
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54 (A,B;C,D) = (A ,B ;C ,D ) Example 2: Show that the following drawing cannot be a true perspective drawing of any piece of railway track. D 4 C 8 B 11 A Solution: The cross ratio (A, B; C, D) on the drawing is 19 8 23 12 = 57 46 . The corresponding four points on the original railway track have cross ratio 2 1 3 2 = 4/3. Since these are different this picture can’t be an accurate picture of the railway track. §4.2. Projective Cross Ratio Theorem 1: The scalar λ in the Collinearity Lemma is unique (ie depends only on the four points and the order in which they are listed.) Proof: Suppose, using the Collinearity Lemma twice, P = p ± = p ± ; Q = q ± = q ± ; R = p + q ± = p + q ± ; S = λ p + q ± = λ′ p + q ± . Then for some scalars α , β , γ , δ : p = α p ; q = β q ; A B C D A B C D
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55 p + q = γ ( p + q ); λ′ p + q = δ ( λ p + q ). Hence α p + β q = γ p + γ q , and since p , q are linearly independent, α = β = γ .
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