Curves and Surfaces Represented by Support Function

Transcription

1 Curves and Surfaces Represented by Support Function Zbyněk Šír Institute of Applied Geometry, JKU Linz (joint research with B. Jüttler and J. Gravesen) The research was supported through grant P17387-N12 of the Austrian Science Fund (FWF).

2 Talk overview Motivation Support function representation and its properties Shapes with polynomial support functions Applications and examples Conclusion Zbyněk Šír: Curves and Surfaces Represented by Support Function 2

3 Motivation The class of NURBS curves/surfaces is not closed under offseting and Minkowski sums. These operations are important for example for molding and collision detection. Zbyněk Šír: Curves and Surfaces Represented by Support Function 3

4 Motivation The class of NURBS curves/surfaces is not closed under offseting and Minkowski sums. These operations are important for example for molding and collision detection. Two solutions to the problem Approximate the result after each operation. Approximate (construct) once for ever the original objects with shapes from a class closed under desired operations. Zbyněk Šír: Curves and Surfaces Represented by Support Function 3-a

5 Motivation The class of NURBS curves/surfaces is not closed under offseting and Minkowski sums. These operations are important for example for molding and collision detection. Two solutions to the problem Approximate the result after each operation. Approximate (construct) once for ever the original objects with shapes from a class closed under desired operations. We look for [Sabin, 1974] a class of curves/surfaces closed under Translation Rotation Scaling Offseting Minkowski sums Zbyněk Šír: Curves and Surfaces Represented by Support Function 3-b

6 Support function representation For any real function h(n) on the unit sphere (or on its subset) h : S n R we define the maping setting x h : S n R n+1 x h (n) = h(n)n + S nh n. We have a linear operator E : h x h : C 1 (S n, R) C 0 (S n, R n+1 ). Zbyněk Šír: Curves and Surfaces Represented by Support Function 4

7 Support function representation n = 1 For any real function h(n) on the unit sphere (or on its subset) PSfrag replacements h : S 1 R we define the maping x h : S 1 R 2 setting }{{} h(n) h (n) x h (n) = h(n)n + h (n)n. }{{} We have a linear operator E : h x h : C 1 (S 1, R) C 0 (S 1, R 2 ). Zbyněk Šír: Curves and Surfaces Represented by Support Function 5

8 Support function representation n = 1 For any real function h(n) on the unit sphere (or on its subset) PSfrag replacements h : S 1 R h (n) }{{} }{{} we define the maping x h : S 1 R 2 h(n) setting x h (n) = h(n)n + h (n)n. We have a linear operator E : h x h : C 1 (S 1, R) C 0 (S 1, R 2 ). Zbyněk Šír: Curves and Surfaces Represented by Support Function 6

9 Support function representation n = 1 For any real function h(n) on the unit sphere (or on its subset) PSfrag replacements h : S 1 R we define the maping x h : S 1 R 2 setting }{{} h(n) h (n) x h (n) = h(n)n + h (n)n. }{{} We have a linear operator E : h x h : C 1 (S 1, R) C 0 (S 1, R 2 ). Zbyněk Šír: Curves and Surfaces Represented by Support Function 7

10 Support function representation n = 1 For any real function h(n) on the unit sphere (or on its subset) PSfrag replacements h : S 1 R we define the maping x h : S 1 R 2 setting }{{} h(n) h (n) x h (n) = h(n)n + h (n)n. }{{} We have a linear operator E : h x h : C 1 (S 1, R) C 0 (S 1, R 2 ). Zbyněk Šír: Curves and Surfaces Represented by Support Function 8

11 Support function representation n = 1 For any real function h(n) on the unit sphere (or on its subset) PSfrag replacements h : S 1 R we define the maping x h : S 1 R 2 setting }{{} h(n) h (n) x h (n) = h(n)n + h (n)n. }{{} We have a linear operator E : h x h : C 1 (S 1, R) C 0 (S 1, R 2 ). Zbyněk Šír: Curves and Surfaces Represented by Support Function 9

12 Support function representation n = 1 For any real function h(n) on the unit sphere (or on its subset) PSfrag replacements h : S 1 R we define the maping x h : S 1 R 2 setting }{{} h(n) h (n) x h (n) = h(n)n + h (n)n. }{{} We have a linear operator E : h x h : C 1 (S 1, R) C 0 (S 1, R 2 ). Zbyněk Šír: Curves and Surfaces Represented by Support Function 10

13 Support function representation n = 1 For any real function h(n) on the unit sphere (or on its subset) PSfrag replacements h : S 1 R we define the maping x h : S 1 R 2 setting }{{} h(n) h (n) x h (n) = h(n)n + h (n)n. }{{} We have a linear operator E : h x h : C 1 (S 1, R) C 0 (S 1, R 2 ). Zbyněk Šír: Curves and Surfaces Represented by Support Function 11

14 Support function representation n = 1 For any real function h(n) on the unit sphere (or on its subset) PSfrag replacements h : S 1 R we define the maping x h : S 1 R 2 setting }{{} h(n) h (n) x h (n) = h(n)n + h (n)n. }{{} We have a linear operator E : h x h : C 1 (S 1, R) C 0 (S 1, R 2 ). Zbyněk Šír: Curves and Surfaces Represented by Support Function 12

15 Support function representation n = 1 For any real function h(n) on the unit sphere (or on its subset) PSfrag replacements h : S 1 R we define the maping x h : S 1 R 2 setting }{{} h(n) h (n) x h (n) = h(n)n + h (n)n. }{{} We have a linear operator E : h x h : C 1 (S 1, R) C 0 (S 1, R 2 ). Zbyněk Šír: Curves and Surfaces Represented by Support Function 13

16 Support function representation n = 1 For any real function h(n) on the unit sphere (or on its subset) PSfrag replacements h : S 1 R we define the maping x h : S 1 R 2 setting }{{} h(n) h (n) x h (n) = h(n)n + h (n)n. }{{} We have a linear operator E : h x h : C 1 (S 1, R) C 0 (S 1, R 2 ). Zbyněk Šír: Curves and Surfaces Represented by Support Function 14

17 Geometric interpretation E : h(n) x h (n) : C 1 (S n, R) C 0 (S n, R n+1 ) }{{} oriented shapes We can interpret x h C 0 (S n, R n+1 ) as an oriented shape, i.e. as the image of x h together normal vectors n attached at x h (n) q h := {[x h (n), n] : n S n }. We will denote Q n the set of all such shapes and call them oriented quasi-convex shapes (curves, surfaces). Q n forms a linear space with respect to Minkowski sum and scalling. E : h q h : support function ( C 1 (S n, R), c, + ) quasiconvex oriented shape (Q n, scaling, Mink. sum ) is a linear operator invariant with respect to action of SO(n + 1). Zbyněk Šír: Curves and Surfaces Represented by Support Function 15

18 Examples Support function Quasi-convex shape const. n v sin(sin(θ)) + cos(cos(θ)) Zbyněk Šír: Curves and Surfaces Represented by Support Function 16

19 Geometric invariance Geometric operation Translation by vector v Scaling by factor c R Offseting with distance d Minkowski sum Rotation by matrix µ SO(n + 1) Change of orientation (all normals reversed) Modified support h v (n) = h(n) + n v h c (n) = c h(n) h d (n) = h(n) + d h(n) = h 1 (n) + h 2 (n) h µ (n) = h(µ 1 (n)) h (n) = h( n) Any subspace of quasi-convex shapes with support functions containing constants and restrictions of linear functions is closed under all desired operations!!! Zbyněk Šír: Curves and Surfaces Represented by Support Function 17

20 Properties of support function representation The quasi-convex shapes have the same (geometrical) smoothness as the support function (with exception for cusps) Sobolev distance of supports is an upper bound on the Hausdorff distance of the shapes max n S n(h 1 h 2 ) 2 + ( S nh 1 S nh 2 ) 2 Hausd(q h1, q h2 ) The Gauß curvature is κ = 1 det (Hess(h) + hi) = 1 h. + h }{{} n=1 Quasi-convex shapes are just offsets of convex shapes. Zbyněk Šír: Curves and Surfaces Represented by Support Function 18

21 Example κ = 1 h + h PSfrag replacements 1 κ θ Zbyněk Šír: Curves and Surfaces Represented by Support Function 19

22 Polynomial support functions We need restrictions of linear functions anyway. Define quasi-convex shape of order k - shape with the support function which is a restriction of a polynomial of degree k on R n+1. Any pseudoconvex shape of order k admits a rational parametrization of degree 2k + 2. Example: h is restriction of x 4 + 2y 4 Is there some nice basis? Zbyněk Šír: Curves and Surfaces Represented by Support Function 20

23 Polynomial support functions h = cos(kθ) = k 2 ( ) k ( 1) l y k 2l (1 y 2 ) l 2l l=0 gives the hypocycloid with circle radii k and k will be called hypocycloid of order k. Y T S α P PSfrag replacements K θ L O Zbyněk Šír: Curves and Surfaces Represented by Support Function 21

24 Basis of quasi-convex curves of finite order (linear space with respect to Mink. sum and scaling) Zbyněk Šír: Curves and Surfaces Represented by Support Function 22

25 Basis of quasi-convex surfaces of finite order Zbyněk Šír: Curves and Surfaces Represented by Support Function 23

26 Application and examples The operator E is linear. It has nice norm properties. The support function express linearly the geometric properties (tangent line, curvature). We have identified suitable sets of support functions giving rational shapes and closed under desired operations. We have nice basis of these sets. Work with polynomial supports = quasi-convex shapes of finite order!! Interpolation (G 1 and G 2 ) Approximation (in L 2 or L or Sobolev norm) Combination of both interpolation and approximation Zbyněk Šír: Curves and Surfaces Represented by Support Function 24

27 Global approximation in L 2 norm 1.5 k = ɛ = mag = Zbyněk Šír: Curves and Surfaces Represented by Support Function 25

28 Global approximation in L 2 norm 1.5 k = ɛ = mag = Zbyněk Šír: Curves and Surfaces Represented by Support Function 25-a

29 Global approximation in L 2 norm 1.5 k = ɛ = mag = Zbyněk Šír: Curves and Surfaces Represented by Support Function 25-b

30 Global approximation in L 2 norm 1.5 k = ɛ = mag = Zbyněk Šír: Curves and Surfaces Represented by Support Function 25-c

31 Global approximation in L 2 norm 1.5 k = ɛ = mag = Zbyněk Šír: Curves and Surfaces Represented by Support Function 25-d

32 Global approximation in L 2 norm 1.5 k = ɛ = mag = Zbyněk Šír: Curves and Surfaces Represented by Support Function 25-e

33 Global approximation in L 2 norm 1.5 k = ɛ = mag = Zbyněk Šír: Curves and Surfaces Represented by Support Function 25-f

34 Global approximation in L 2 norm 1.5 k = ɛ = mag = Zbyněk Šír: Curves and Surfaces Represented by Support Function 25-g

35 Global approximation in L 2 norm 1.5 k = ɛ = mag = Zbyněk Šír: Curves and Surfaces Represented by Support Function 25-h

36 Global approximation in L 2 norm Ellipsoid with half-axes 1, 2 and 2 has support function, which is restriction of x2 + 2y 2 + 4z 2 defined on R 3. Approximation by polynomials up to degree 6 has error Zbyněk Šír: Curves and Surfaces Represented by Support Function 26

37 Piecewise constructions = Zbyněk Šír: Curves and Surfaces Represented by Support Function 27

38 Piecewise constructions 2 Zbyněk Šír: Curves and Surfaces Represented by Support Function 28

39 Piecewise constructions 2 Zbyněk Šír: Curves and Surfaces Represented by Support Function 28-a

40 Piecewise constructions 2 Zbyněk Šír: Curves and Surfaces Represented by Support Function 28-b

41 Piecewise constructions 2 Zbyněk Šír: Curves and Surfaces Represented by Support Function 28-c

42 Piecewise constructions 2 Zbyněk Šír: Curves and Surfaces Represented by Support Function 28-d

43 Piecewise constructions = Zbyněk Šír: Curves and Surfaces Represented by Support Function 29

44 Conclusion We have identified a class of rational pseudoconvex curves/surfaces which is closed with respect to offseting and Minkowski sums. The geometric and metric properties of these shapes are controlled efficiently by the support function. We have found a relatively simple basis for these shapes. We have designed various approximation/interpolation methods based on pseudoconvex shapes. Future work: treatment of inflections and parabolic points. Zbyněk Šír: Curves and Surfaces Represented by Support Function 30

45 Thank you for your attention!