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

This preview shows pages 1–4. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern