Invariant Measures of Convex Sets. Eitan Grinspun, Columbia University Peter Schröder, Caltech

Transcription

1 Invariant Measures of Convex Sets Eitan Grinspun, Columbia University Peter Schröder, Caltech

2 What will we measure? Subject to be measured, S an object living in n-dim space convex, compact subset of R n e.g., points, edges, faces?

3 What will we measure? Subject to be measured, S an object living in n-dim space convex, compact subset of R n CLOSED + BOUNDED e.g., points, edges, faces?

4 What will we measure? Subject to be measured, S an object living in n-dim space convex, compact subset of R n + finite unions and intersections e.g., points, edges, faces a mesh

5 What is a reasonable measure? Properties a measure is scalar-valued empty set additivity normalization (parallelotope, P)

6 What is a reasonable measure? real number, coordinate-frame invariant Properties a measure is scalar-valued empty set additivity normalization (parallelotope, P)

7 What is a reasonable measure? real number, coordinate-frame invariant Properties a measure is scalar-valued empty set additivity normalization (parallelotope, P) example: volume Other measures?

8 Other Measures Elementary symmetric functions area/2 length/4

9 Invariant Measures Intrinsic volumes n measures in n dimensions how to generalize to compact convex sets? Geometric probability measure points in set probability of hitting set

10 Geometric probability Blindly throw darts count number of hits Darts: k-dim subspaces of n-d points lines planes volumes k=0 n=2

11 Geometric probability Indicator function, input: a dart, ω output (point dart): 1 if dart hits body 0 if dart misses body k=0 n=

12 Geometric probability Indicator function, input: a dart, ω output (point dart): 1 if dart hits body 0 if dart misses body in general, output is # hits k=1 n=2k=0 n=

13 Geometric probability Throw N random darts to estimate area R C k=0 n=2

14 Geometric probability Throw N random darts to estimate area Throw all the darts you have

15 Geometric probability Throw N random darts to estimate area Throw all the darts you have 2 nd intrinsic volume of C measure of darts (k=0,n=2) hitting target C

16 Example: lines in R 3 Measure of lines through rectangle, prop. to area of R two ways to explain consider each fixed line orientation in turn consider each point on the rectangle in turn

17 Example: lines in R 3 Additivity in action consider the union of two rectangles

18 Example: lines in R 3 Additivity in action consider the union of two rectangles

19 Example: lines in R 3 Additivity in action consider the union of two rectangles

20 Example: lines in R 3 Additivity in action consider the union of two rectangles

21 Example: lines in R 3 Additivity in action consider the union of two rectangles inclusion - exclusion principle

22 Example: planes in R 3 Measure of planes along a curve, consider simplest curve: line segment proportional to length

23 Example: planes in R 3 Measure of planes along a curve, consider simplest curve: line segment proportional to length

24 Example: planes in R 3 Measure of planes along a curve, consider polyline proportional to length

25 Example: planes in R 3 Measure of planes hitting parallelotope, consider particularly useful polyline indicator fn. for c = indicator fn. for P c y P z x

26 Example: planes in R 3 Measure of planes hitting parallelotope, consider particularly useful polyline indicator fn. for c = indicator fn. for P proportional to length of polyline x 2 x 3 x 1

27 Example: planes in R 3 Measure of planes hitting parallelotope, consider particularly useful polyline indicator fn. for c = indicator fn. for P proportional to length of polyline for general shape E, x 2 x 3 x 1

28 Recap Axioms it is a measure Euclidian motion invariance additivity some normalization µ k (C) for compact convex set is the measure of all linear varieties of dim. n-k meeting C

29 But wait Do we have all of them? there is one missing recall: elementary symmetric functions

30 But wait Do we have all of them? there is one missing recall: elementary symmetric functions what about e 0 (x 1,x 2,,x n )?

31 Euler characteristic Symmetric function of order zero for any compact, convex set: Euler characteristic is this a measure?

32 Euler characteristic Symmetric function of order zero for any compact, convex set: homework! Euler characteristic χ is this a measure?

33 Euler characteristic Is it additive? χ(e 1 ) = 1 e 2 χ(e 2 ) = 1 e 1

34 Euler characteristic Is it additive? χ(e 1 ) = 1 e 1 e 2 e 2 χ(e 2 ) = 1 e 1

35 Euler characteristic Is it additive? χ(e 1 ) = 1 e 1 e 1 e 2 χ(e 1 e 2 ) = 1 e 1 e 2 e 2 χ(e 2 ) = 1

36 Euler characteristic Is it additive? χ(e 1 ) = 1 e 1 e 2 e 2 χ(e 2 ) = 1 e 1 χ(e 1 e 2 ) = = 1

37 Euler characteristic Is it additive? yes Is it invariant under rigid-body motion? yes

38 Continuity Our final axiom measure should be continuous

39 Example: lines in R 3 Measure of lines through planar surface limit process gives surface area for arbitrary planar surface (not just for R)

40 Example: lines in R 3 Measure of lines through planar surface limit process gives surface area for arbitrary planar surface (not just for R) disjoint union of rectangles

41 Example: lines in R 3 Measure of lines through planar surface limit process gives surface area for arbitrary planar surface (not just for R) disjoint union of rectangles

42 Example: lines in R 3 Area of a (non-planar, disjoint) surface, D 1. partition into planar regions, C i Çáëàçáåí=ìåáçå

43 Example: lines in R 3 Area of a (non-planar, disjoint) surface, D 1. partition into planar regions, C i 2. add areas of each region åìãäéê=çñ=íáãéë=ω=ãééíë=a

44 Example: lines in R 3 Area of a (non-planar, disjoint) surface, D 1. partition into planar regions, C i 2. add areas of each region åìãäéê=çñ=íáãéë=ω=ãééíë=a

45 Here s the clincher

46 Hadwiger (1957) These measures form a basis for all continuous, additive, rigid motion invariant measures on ring of convex sets.

47 Hadwiger (1957) These measures form a basis for all continuous, additive, rigid motion invariant measures on ring of convex sets. FUNDAMENAL RESULT

48 Putting it together Steiner, Cauchy, Hadwiger expand a convex set outward by epsilon ãáå=çáëí~ååé=íç=ëéí

49 Putting it together Steiner, Cauchy, Hadwiger expand a convex set outward by epsilon ãáå=çáëí~ååé=íç=ëéí the increase in volume is a polynomial in ε the coefficients are the intrinsic volumes!

50 Steiner example in R 3 volume area mean width Euler charac.

51 Steiner example in R 3 volume area mean width Euler charac. Rewrite as boundary integral: symmetric even f n of principal curvatures: H 0 =1, H 1 =(κ 1 +κ 2 ), H 2 =κ 1 κ 2

52 Steiner example in R 3 Rewrite as boundary integral: symmetric even f n of principal curvatures: H 0 =1, H 1 =(κ 1 +κ 2 ), H 2 =κ 1 κ 2

53 Steiner example in R 3 = surface area Rewrite as boundary integral: symmetric even f n of principal curvatures: H 0 =1, H 1 =(κ 1 +κ 2 ), H 2 =κ 1 κ 2

54 Steiner example in R 3 = surface area Rewrite as boundary integral: symmetric even f n of principal curvatures: H 0 =1, H 1 =(κ 1 +κ 2 ), H 2 =κ 1 κ 2

55 Steiner example in R 3 = surface area = total mean curvature Rewrite as boundary integral: symmetric even f n of principal curvatures: H 0 =1, H 1 =(κ 1 +κ 2 ), H 2 =κ 1 κ 2

56 Steiner example in R 3 = surface area = total mean curvature` Rewrite as boundary integral: symmetric even f n of principal curvatures: H 0 =1, H 1 =(κ 1 +κ 2 ), H 2 =κ 1 κ 2

57 Steiner example in R 3 = surface area = total mean curvature` Rewrite as boundary integral: symmetric even f n of principal curvatures: H 0 =1, H 1 =(κ 1 +κ 2 ), H 2 =κ 1 κ 2 = Euler charac.

58 A Steiner walk-through, 2d Inflate a planar polygon by epsilon A What is the new area?

59 A Steiner walk-through, 2d Each edge contributes a rectangle

60 A Steiner walk-through, 2d Each vertex contributes a sector

61 A Steiner walk-through, 3d Inflate a polyhedron What is the new volume?

62 A Steiner walk-through, 3d Each face contributes a parallelotope

63 A Steiner walk-through, 3d Each edge contributes a wedge of a cylinder

64 A Steiner walk-through, 3d Each vertex contributes a spherical wedge

65 A Steiner walk-through, 3d α Each vertex contributes a spherical wedge

66 A Steiner walk-through, 3d

67 A Steiner walk-through, 3d

68 A Steiner walk-through, 3d

69 A Steiner walk-through, 3d

70 A Steiner walk-through, 3d

71 How to use? Want to measure deformation? only very few measures need apply volume, area, mean width, χ Define curvatures of polyhedra? deal with convexity issue kçêã~ä=åóåäé `çüéåjpíéáåéêljçêî~å