Optimal (local) Triangulation of Hyperbolic Paraboloids

Transcription

1 Optimal (local) Triangulation of Hyperbolic Paraboloids Dror Atariah Günter Rote Freie Universität Berlin December 14 th 2012

2 Outline Introduction Taylor Expansion Quadratic Surfaces Vertical Distance Bounding the Hausdorff Distance You look familiar... Back to the plane Saddle in Normal Form z = xy General Saddles Conclusion Saddles Triangulation: Introduction 2 / 29

3 Motivation Ultimate goal Generate optimal triangulation of patches of smooth surfaces. Current goal Generate a local optimal triangulation of patches of smooth surfaces. Current road map Obtain smooth local approximation of a given surface. Obtain optimal triangulation of the approximation. Deduce an optimal local triangulation of the original surface. Saddles Triangulation: Introduction 3 / 29

4 Local Approximation of smooth surfaces Taylor expansion of a smooth surface S about a point p up to second order terms yields a quadratic surface (up to Euclidean transformation). Depending on the principal curvatures at p the point is either elliptic or hyperbolic or parabolic or planar. The approximating patch corresponds the point s type. Saddles Triangulation: Introduction: Taylor Expansion 4 / 29

5 Taylor expansion of a smooth surface S about a point p up to second order terms yields a quadratic surface (up to Euclidean transformation). Depending on the principal curvatures at p the point is either parabolic or hyperbolic or parabolic or planar. The approximating patch corresponds the point s type. Z Local Approximation of smooth surfaces Y X Saddles Triangulation: Introduction: Taylor Expansion 4 / 29

6 Taylor expansion of a smooth surface S about a point p up to second order terms yields a quadratic surface (up to Euclidean transformation). Depending on the principal curvatures at p the point is either elliptic or hyperbolic or parabolic or planar. The approximating patch corresponds the point s type. Z Local Approximation of smooth surfaces Y X Saddles Triangulation: Introduction: Taylor Expansion 4 / 29

7 Taylor expansion of a smooth surface S about a point p up to second order terms yields a quadratic surface (up to Euclidean transformation). Depending on the principal curvatures at p the point is either elliptic or hyperbolic or parabolic or planar. The approximating patch corresponds the point s type. Z Local Approximation of smooth surfaces Y X Saddles Triangulation: Introduction: Taylor Expansion 4 / 29

8 Taylor expansion of a smooth surface S about a point p up to second order terms yields a quadratic surface (up to Euclidean transformation). Depending on the principal curvatures at p the point is either elliptic or hyperbolic or parabolic or planar. The approximating patch corresponds the point s type. Z Local Approximation of smooth surfaces Y X Saddles Triangulation: Introduction: Taylor Expansion 4 / 29

9 Outline Introduction Taylor Expansion Quadratic Surfaces Vertical Distance Bounding the Hausdorff Distance You look familiar... Back to the plane Saddle in Normal Form z = xy General Saddles Conclusion Saddles Triangulation: Quadratic Surfaces 5 / 29

10 Quadratic Surface Definition The zero set of a polynomial in three variables of degree at most two is called a quadratic surface. to sy rte e cu shir s y e b ag er Im m D dia. Sa kipe i W Ellipsoid Paraboloid Hyperboloid Hyperboloid Hyp. Paraboloid Elliptical Cone Saddles Triangulation: Quadratic Surfaces 6 / 29

11 Quadratic Surface Definition The zero set of a polynomial in three variables of degree at most two is called a quadratic surface. to sy rte e cu shir s y e b ag er Im m D dia. Sa kipe i W Ellipsoid Paraboloid Hyperboloid Hyperboloid Hyp. Paraboloid Elliptical Cone Saddles Triangulation: Quadratic Surfaces 6 / 29

12 Our Surfaces For the following polynomial F(x y) = a11 x2 + a12 xy + a22 y2 + a13 x + a23 y + a33 we will consider the corresponding graph namely S = (x y z) R3 : z = F(x y). Z Z Z Depending on the coefficients the resulting surface can be either a plane or an elliptic paraboloid or a hyperbolic paraboloid (saddle) or a parabolic cylinder. Y X Saddles Triangulation: Quadratic Surfaces Y Y X X 7 / 29

13 Summary Locally using Taylor expansion every smooth surface is approximated by a quadratic patch. Using Euclidean motions the quadratic patches can be transformed to graphs of bi-variate polynomials. So let us approximate quadratic graphs! Let s concentrate on saddles... Saddles Triangulation: Quadratic Surfaces 8 / 29

14 Outline Introduction Taylor Expansion Quadratic Surfaces Vertical Distance Bounding the Hausdorff Distance You look familiar... Back to the plane Saddle in Normal Form z = xy General Saddles Conclusion Saddles Triangulation: Vertical Distance 9 / 29

15 The Vertical Distance Before talking about approximations we have to define the error measure that we will use. Definition (Vertical Distance) Given a compact domain D R2 and two functions f g: D R then the vertical distance between f and g is distv (f g) = max f (x y) g(x y). (xy) D Why using the Vertical Distance? Saddles Triangulation: Vertical Distance 10 / 29

16 Vertical and Hausdorff Distances Lemma Let A B R3 be two sets such that their projection to the plane is identical. Then the following holds disth (A B) distv (A B) Proof. For all a A we have: dist (a B) = inf { dist (a b) : b B } dist a Π 1 (a) distv (A B) Thus dist (A B) = sup { dist (a B) : a A } distv (A B) Saddles Triangulation: Vertical Distance: Bounding the Hausdorff Distance 11 / 29

17 The Bad News... This bound depends on the slope and can be arbitrarily bad. v α α h The following holds: cos α = h v Saddles Triangulation: Vertical Distance: Bounding the Hausdorff Distance 12 / 29

18 Loosing the Generality or you look familiar... Lemma For every point p S there exists an affine transformation Tp : R3 R3 which satisfies the following: It maps the point p to the origin. It maps S to a quadratic graph surface S which is given by a polynomial of the form F (x y) = a1 x2 + 2a2 xy + a3 y2. Finally for any pair of points q r R3 which lie on a vertical line then we have q r = T (q) T (r). Corollary We can triangulate the vicinity of the origin of the quadratic graph. Saddles Triangulation: Vertical Distance: You look familiar / 29

19 Vertical distance from a line to the graph Lemma Given two points p q on a quadratic graph S then 1 distv ℓpq S = F(px qx py qy ) 4 where: ℓpq is the line segment connecting p and q. F(x y) is the bi-variate polynomial which defines S. Y X Saddles Triangulation: Vertical Distance: Back to the plane 14 / 29

20 Interpolating Triangle What about the vertical distance from an interpolating triangle with vertices p0 p1 and p2 to a saddle S? p1 p0 p2 Y X Saddles Triangulation: Vertical Distance: Back to the plane 15 / 29

21 Condition for Optimality Lemma A triangle T with vertices p0 p1 p2 S such that pi = (xi yi zi ) satisfies distv (T S) ε if and only if all its edge vectors satisfy 1 4 F(xi xj yi yj ) ε Saddles Triangulation: Vertical Distance: Back to the plane 16 / 29

22 Outline Introduction Taylor Expansion Quadratic Surfaces Vertical Distance Bounding the Hausdorff Distance You look familiar... Back to the plane Saddle in Normal Form z = xy General Saddles Conclusion Saddles Triangulation: Saddle in Normal Form z = xy 17 / 29

23 Find an Optimal Triangle Goal Find a triangle T with vertices p0 p1 p2 S of maximal area such that distv (T S) = ε In practice... In practice find T of maximal area such that distv (T S) ε Saddles Triangulation: Saddle in Normal Form z = xy 18 / 29

24 Find an Optimal Triangle - Step by Step Recall Assume that p0 = (0 0 0). Find p1 p2 where pi = (xi yi zi ) such that x1 y1 ε (x1 x2 )(y1 y2 ) ε to assure distv (T S) ε. Saddles Triangulation: Saddle in Normal Form z = xy 19 / 29

25 Find an Optimal Triangle - Step by Step (cont.) y p0 x y x = ε Saddles Triangulation: Saddle in Normal Form z = xy 20 / 29

26 Find an Optimal Triangle - Step by Step (cont.) y p1 = (ξ ξε ) e~0 p0 x y x = ε Saddles Triangulation: Saddle in Normal Form z = xy 20 / 29

27 Find an Optimal Triangle - Step by Step (cont.) (x y ξ) y = ε p1 = (ξ ξε ) e~0 p0 x y x = ε Saddles Triangulation: Saddle in Normal Form z = xy 20 / 29

28 Find an Optimal Triangle - Step by Step (cont.) (x y ξ) y = ε p1 = (ξ ξε ) e~0 p0 x y x = ε Saddles Triangulation: Saddle in Normal Form z = xy 20 / 29

29 Find an Optimal Triangle - Step by Step (cont.) (x y ξ) y = ε p1 = (ξ ξε ) e~0 p0 x y x = ε Saddles Triangulation: Saddle in Normal Form z = xy 20 / 29

30 Find an Optimal Triangle - Step by Step (cont.) (x y ξ) y p23 = ε p1 = (ξ ξε ) p22 e~0 p21 p0 p26 x p25 x y p24 = ε Saddles Triangulation: Saddle in Normal Form z = xy 20 / 29

31 Find an Optimal Triangle - Step by Step (cont.) (x y ξ) y p23 = ε p1 = (ξ ξε ) p22 e~0 p0 T4 (ξ) y x e~1 e~2 p21 p26 x p25 p24 = ε This construction yields six one-parameter families of triangles! All the triangles have the same maximal area! Saddles Triangulation: Saddle in Normal Form z = xy 20 / 29

32 Make them nicer! Goal Maximize the minimal angle! p23 y y p22 p1 = (ξ ξε ) 1 T1 ( ϕ ) 1 T6 ( ϕ ) p21 p26 x x p0 p25 1 T6 ( ϕ ) 1 T1 ( ϕ ) p24 Saddles Triangulation: Saddle in Normal Form z = xy 21 / 29

33 Make them nicer! Goal Maximize the minimal angle! y y p23 p22 T2 (1) p1 = (ξ ξε ) p21 p26 x p0 p25 p24 T5 ( 1) T5 (1) x T2 ( 1) Saddles Triangulation: Saddle in Normal Form z = xy 21 / 29

34 Make them nicer! Goal Maximize the minimal angle! y y p23 T3 (ϕ) p1 = (ξ ξε ) p22 p21 x p0 p26 T4 ( ϕ) p25 T4 (ϕ) x p24 T3 ( ϕ) Saddles Triangulation: Saddle in Normal Form z = xy 21 / 29

35 Make them nicer! Goal Maximize the minimal angle! y 1 T1 ( ϕ ) T3 (ϕ) T4 ( ϕ) T3 ( ϕ) T4 (ϕ) x 1 T1 ( ϕ ) Saddles Triangulation: Saddle in Normal Form z = xy 21 / 29

36 Make them nicer! Goal Maximize the minimal angle! y T3 (ϕ) Rotating by 90 T4 (ϕ) 1 T1 ( ϕ ) 1 T1 ( ϕ ) T4 ( ϕ) x T3 ( ϕ) Saddles Triangulation: Saddle in Normal Form z = xy 21 / 29

37 Make them nicer! Goal Maximize the minimal angle! y y p23 p22 p1 = (ξ ξε ) p21 T2 (1) p26 x p0 p25 T5 ( 1) T5 (1) x p24 T2 ( 1) The conjugate case! Saddles Triangulation: Saddle in Normal Form z = xy 21 / 29

38 Back to Space! Project the planar triangulation to the surface y x y x Saddles Triangulation: Saddle in Normal Form z = xy 22 / 29

39 Summary We obtained a triangulation of the vicinity of the origin. All the triangles have a vertical distance equals to ε from the saddle. Hausdorff distance is at most ε. Saddles Triangulation: Saddle in Normal Form z = xy 23 / 29

40 From Local to Global Can we triangulate a surface globally? Saddles Triangulation: Saddle in Normal Form z = xy 24 / 29

41 From Local to Global Can we triangulate a surface globally? We can with the saddle... Saddles Triangulation: Saddle in Normal Form z = xy 24 / 29

42 From Local to Global Can we triangulate a surface globally? We can with the saddle... The optimal triangulation is invariant under translations thus we can tesselate the plane! Saddles Triangulation: Saddle in Normal Form z = xy 24 / 29

43 From Local to Global Can we triangulate a surface globally? We can with the saddle... The optimal triangulation is invariant under translations thus we can tesselate the plane!... and can be projected to the saddle. Saddles Triangulation: Saddle in Normal Form z = xy 24 / 29

44 Outline Introduction Taylor Expansion Quadratic Surfaces Vertical Distance Bounding the Hausdorff Distance You look familiar... Back to the plane Saddle in Normal Form z = xy General Saddles Conclusion Saddles Triangulation: General Saddles 25 / 29

45 Triangulating the General Saddles A(n) (almost) general saddle is «x2 y2 (x y z) : 2 2 = z a b Saddles Triangulation: General Saddles 26 / 29

46 Triangulating the General Saddles Hyperbola A(n) (almost) general saddle is «x2 y2 (x y z) : 2 2 = z a b So far we didn t care about conjugate cases. its conjugate Saddles Triangulation: General Saddles 26 / 29

47 Triangulating the General Saddles Hyperbola A(n) (almost) general saddle is «x2 y2 (x y z) : 2 2 = z a b So far we didn t care about conjugate cases. This changes in the case of a general saddle. its conjugate Saddles Triangulation: General Saddles 26 / 29

48 Finding Optimal Planar Triangles There and back again... Saddles Triangulation: General Saddles 27 / 29

49 Finding Optimal Planar Triangles There... Anisotropic scale... and back again... Saddles Triangulation: General Saddles 27 / 29

50 Finding Optimal Planar Triangles There... Anisotropic scale Obtain right hyperbola... and back again... Saddles Triangulation: General Saddles 27 / 29

51 Finding Optimal Planar Triangles There... Anisotropic scale Obtain right hyperbola 45 rotation will yield the case we studied... and back again... Saddles Triangulation: General Saddles 27 / 29

52 Finding Optimal Planar Triangles There... Anisotropic scale Obtain right hyperbola 45 rotation will yield the case we studied Triangulate the rectangular case... and back again... y x y x Saddles Triangulation: General Saddles 27 / 29

53 Finding Optimal Planar Triangles There... Anisotropic scale Obtain right hyperbola 45 rotation will yield the case we studied Triangulate the rectangular case... and back again... Map the triangulation back to the general hyperbola. We conjecture that one image is optimal. y x y x Saddles Triangulation: General Saddles 27 / 29

54 Finding Optimal Planar Triangles There... Anisotropic scale Obtain right hyperbola 45 rotation will yield the case we studied Triangulate the rectangular case... and back again... Map the triangulation back to the general hyperbola. We conjecture that one image is optimal. and to the saddle! y x y x Saddles Triangulation: General Saddles 27 / 29

55 Outline Introduction Taylor Expansion Quadratic Surfaces Vertical Distance Bounding the Hausdorff Distance You look familiar... Back to the plane Saddle in Normal Form z = xy General Saddles Conclusion Saddles Triangulation: Conclusion 28 / 29

56 Time Line Previous work On piecewise linear approximation of quadratic functions Pottmann et al (Journal for Geometry and Graphics 00 ) What s next? Study the local approximation further (e.g. Discrete Gaussian curvature). Use the local approximation to obtain global approximation of smooth surfaces. Saddles Triangulation: Conclusion 29 / 29

57 Time Line Previous work On piecewise linear approximation of quadratic functions Pottmann et al (Journal for Geometry and Graphics 00 ) What s next? Study the local approximation further (e.g. Discrete Gaussian curvature). Use the local approximation to obtain global approximation of smooth surfaces. Thank you for your attention! dror.atariah@fu-berlin.de Saddles Triangulation: Conclusion 29 / 29