Understanding Projective Invariants in Numerical Work

1. Numerical Projective Invariants

2. FUMIKO'S WORK

3. Equally Spaced Points

4. (one end) ÷ (middle) •(other end) ÷ (total)

6. Some Notation Given distinct points A, B, we denote

7. Given four collinear points A, B, C, D, regardless of order, we define the cross ratio X(ABCD) by where ||*|| denotes directed distance.

9. The absolute value is very useful, too:

11. Rearranging the labels gives a different value:

12. Given four collinear points, and any arrangement of the labels A, B, C, D, the cross ratio X(ABCD) is projectively invariant. Naturally, the same is true of |X(ABCD)|.

13. That is, if A, B, C, D are collinear points, and A', B', C', D' are their corresponding projective images, then and

15. When one point is at infinity, (an ideal point), we formally compute |x(ABCD)| as follows:

16. In the plan view (unprimed letters),

17. Thus by the invariance of the cross ratio, (a harmonic range)

18. Recall that the image of a straight line not parallel to the picture plane has a vanishing point.

20. An application (see worksheet): l l

21. image not a right angle

22. Projective geometry: a geometry “which disregards all considerations of distance and angle.” H. S. M. Coxeter and S. L. Greitzer Geometry Revisited MAA, Washington, D.C., 1967

29. HOMEWORK For more information, see the paper A Different Angle on Perspective The College Mathematics Journal Vol. 43, No. 5 (2012), 354–360.

31. F G E H D A B C

32. Definition. A product of ratios of (either directed or ordinary) distances, where all the indicated points lie in one plane, is called an h-expression if it has the following properties: • In each ratio the points that occur are collinear. • Each point appears in the numerator of the product exactly as many times as it does in the denominator.

33. Theorem (Howard W. Eves, 1913–2000) The value of an h-expression is invariant under any projective transformation.

34. Eves's theorem can even be generalized to non-planar polygons. Here is invariant!

35. The famous cross ratio is an h-expression: Its projective invariance is a special case of Eves’ theorem!

36. “We feel that Eves’ theorem has never been given the recognition it deserves and should be regarded as one of the fundamental results of projective geometry.” G. C. Shephard

37. Eves’ theorem appeared with little fanfare in his textbook A Survey of Geometry Allyn and Bacon, Boston (1963)

38. HOW FAST WAS THIS CAR GOING?

39. THE KEY STEP IS THE DETERMINATION OF THE SKID MARK LENGTH |AB|. THE WHITE CAR (1969 DODGE CHARGER)

40. IT’S EASY WITH EVES’ THEOREM: |AB| =(WHEELBASE) × (h-EXPRESSION). LOCATED ARBITRARILY I MEASURED, AND CALCULATED ≈ 1.5

41. MORE HOMEWORK See the paper A Car Crash Solved—with A Swiss Army Knife Mathematics Magazine Vol. 84, No. 5, 2011, 327–338

42. Alternate solution: when D is infinite, 1969 Dodge Charger

43. Thus, with the vanishing point D', Or equivalently,

44. HOWARD W. EVES 1911–2004