(Improved) Optimal Triangulation of Saddle Surfaces Computational - PowerPoint PPT Presentation

(Improved) Optimal Triangulation of Saddle Surfaces Computational Geometric Learning (CGL) supported by EU FET-Open grant Transregio-SFB Discretization in Geometry and Dynamics (DGD) D. Atariah G. Rote M. Wintraecken Freie Universität Berlin, Rijksuniversiteit Groningen SFB DGD Workshop, Schloss Schley, November 2013

Motivation ◮ Smooth surface is locally approximated by a quadratic patch. ◮ Euclidean motion transforms the quadratic patch to graph of a bi-variate polynomial. ◮ → approximate graphs of quadratic polynomials! � ( x, y, z ) : z = F ( x, y ) � • H. Pottmann, R. Krasauskas, B. Hamann, K. Joy, and W. Seibold: On piecewise linear approximation of quadratic functions. Journal for Geometry and Graphics 4 (2000), 31–53. , Optimal Triangulation ➢ Introduction SFB DGD Workshop 2

Outline Introduction Interpolating Approximation Non-interpolating Approximation , Optimal Triangulation ➢ Introduction SFB DGD Workshop 3

Vertical Distance ◮ We are interested in a neighborhood of some point. ◮ Make the surface normal vertical. ◮ The direction in which Hausdorff distance is measured becomes almost vertical. Definition (Vertical Distance, L ∞ Distance) Given two domains D 1 , D 2 ⊂ R 2 and two graphs f : D 1 → R and g : D 2 → R then the vertical distance is dist V ( f, g ) = max | f ( x, y ) − g ( x, y ) | ( x,y ) ∈ D 1 ∩ D 2 , Optimal Triangulation ➢ Introduction SFB DGD Workshop 4

Properties of V-Distance Lemma Let A, B ⊂ R 3 be two sets with equal v projection to the plane. Then α dist H ( A, B ) ≤ dist V ( A, B ) α h , Optimal Triangulation ➢ Introduction SFB DGD Workshop 5

V-Distance of Quadratic Functions Lemma (Every two points are the same) Let S be the graph of a quadratic function. For every point p ∈ S, there is an affine transformation T p : R 3 → R 3 which satisfies the following: ◮ T p ( p ) = � 0 ◮ T p ( S ) = a quadratic graph ˜ S with a homogeneous polynomial of the form F ( x , y ) = ax 2 + bxy + cy 2 ˜ ( ∗ ) ◮ For all q , r ∈ R 3 on a vertical line, | q − r | = | T p ( q ) − T p ( r ) | . ◮ T p ( p ) on the first two coordinates is a translation in R 2 . , Optimal Triangulation ➢ Introduction SFB DGD Workshop 6

Vertical Distance of a Chord If S is negatively curved, the maximum distance to a triangle never occurs in the interior. Lemma For a line segment pq between two points p = ( p x , p y , p z ) and q = ( q x , q y , q z ) on a quadratic graph S, dist V ( pq , S ) = 1 � ˜ � � F ( q x − p x , q y − p y ) � 4 ◮ ˜ F ( x , y ) is the homogeneous polynomial ( ∗ ) . ◮ The max. vertical distance is attained at the midpoint. q p , Optimal Triangulation ➢ Introduction SFB DGD Workshop 7

Setup From now on, S = � ( x , y , z ) : z = xy � (by a linear transformation of the x - y -plane) Goal Given ϵ > 0, find a triangle T with vertices p 0 , p 1 , p 2 ∈ S of largest area such that dist V ( T , S ) ≤ ϵ Translated and reflected copies of T have the same error and tile the plane: max. AREA ⇔ min. NUMBER of triangles , Optimal Triangulation ➢ Interpolating Approximation SFB DGD Workshop 8

Maximize the Area of Planar Triangles p 0 | x y | = 4 ϵ , Optimal Triangulation ➢ Interpolating Approximation SFB DGD Workshop 9

Maximize the Area of Planar Triangles p 1 � e 0 p 0 | x y | = 4 ϵ , Optimal Triangulation ➢ Interpolating Approximation SFB DGD Workshop 9

Maximize the Area of Planar Triangles | ( x − ξ ) y | = 4 ϵ p 1 � e 0 p 0 | x y | = 4 ϵ , Optimal Triangulation ➢ Interpolating Approximation SFB DGD Workshop 9

Maximize the Area of Planar Triangles | ( x − ξ ) y | p 2 , 3 = 4 ϵ p 2 , 2 p 1 � p 2 , 1 e 0 p 2 , 6 p 0 p 2 , 5 | x p 2 , 4 y | = 4 ϵ , Optimal Triangulation ➢ Interpolating Approximation SFB DGD Workshop 9

Maximize the Area of Planar Triangles | ( x − ξ ) y | p 2 , 3 = 4 ϵ p 2 , 2 p 1 e 2 � � p 2 , 1 e 0 p 2 , 6 T 4 ( ξ ) p 0 p 2 , 5 e 1 � | x p 2 , 4 y | = 4 ϵ , Optimal Triangulation ➢ Interpolating Approximation SFB DGD Workshop 9

Optimize the Shape of Planar Triangles Secondary criterion: Maximize the smallest angle y x , Optimal Triangulation ➢ Interpolating Approximation SFB DGD Workshop 10

Optimize the Shape of Planar Triangles Secondary criterion: Maximize the smallest angle y x , Optimal Triangulation ➢ Interpolating Approximation SFB DGD Workshop 10

Triangulate the Saddle Lift the planar triangulation to the surface y x y x , Optimal Triangulation ➢ Interpolating Approximation SFB DGD Workshop 11

Can We Do Better? What do we have? Given an ϵ > 0 and a saddle surface S , we can find a family T of triangles which interpolate the surface and ◮ have maximum area, ◮ maintain dist V ( S , T ) ≤ ϵ for all T ∈ T . , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 12

Can We Do Better? What do we have? Given an ϵ > 0 and a saddle surface S , we can find a family T of triangles which interpolate the surface and ◮ have maximum area, ◮ maintain dist V ( S , T ) ≤ ϵ for all T ∈ T . Question. . . ◮ Can this be improved by allowing non-interpolating triangles? ◮ Pottmann et al. (2000) conjectured NO. This question is easy for convex approximation. , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 12

Pseudo-Euclidean Transformations ◮ A λ -pseudo Euclidean map is given by: ( x , y ) �→ ( λ x , 1 λ y ) ◮ Vertical distance is preserved. ◮ Area (projected) is preserved. ◮ Surface S = { z = xy } is preserved. , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 13

Pseudo-Euclidean Transformations ◮ A λ -pseudo Euclidean map is p 2 given by: ( x , y ) �→ ( λ x , 1 λ y ) ◮ Vertical distance is preserved. ◮ Area (projected) is preserved. p 1 ◮ Surface S = { z = xy } is preserved. p 0 , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 13

Pseudo-Euclidean Transformations ◮ A λ -pseudo Euclidean map is given by: p 2 ( x , y ) �→ ( λ x , 1 λ y ) ◮ Vertical distance is preserved. ◮ Area (projected) is preserved. p 1 ◮ Surface S = { z = xy } is preserved. p 0 , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 13

Pseudo-Euclidean Transformations ◮ A λ -pseudo Euclidean map is given by: p 2 ( x , y ) �→ ( λ x , 1 λ y ) ◮ Vertical distance is preserved. ◮ Area (projected) is preserved. p 1 ◮ Surface S = { z = xy } is preserved. p 0 , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 13

Pseudo-Euclidean Transformations ◮ A λ -pseudo Euclidean map is given by: p 2 ( x , y ) �→ ( λ x , 1 λ y ) ◮ Vertical distance is preserved. ◮ Area (projected) is preserved. p 1 ◮ Surface S = { z = xy } is preserved. p 0 , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 13

Pseudo-Euclidean Transformations ◮ A λ -pseudo Euclidean map is given by: ( x , y ) �→ ( λ x , 1 λ y ) p 2 ◮ Vertical distance is preserved. ◮ Area (projected) is preserved. p 1 ◮ Surface S = { z = xy } is preserved. p 0 , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 13

In the Plane Fact The area of the (interpolating) optimal triangles in the plane is � 2 5 ϵ . below S above S , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 14

In the Plane Fact The area of the (interpolating) optimal triangles in the plane is � 2 5 ϵ . p 2 = ( η, ξ ) p 1 = ( ξ, η ) p 0 , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 14

In the Plane Fact The area of the (interpolating) optimal triangles in the plane is � 2 5 ϵ . p 2 = ( η, ξ ) p 1 = ( ξ, η ) p 0 , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 14

In the Plane Fact The area of the (interpolating) optimal triangles in the plane is � 2 5 ϵ . p 2 = ( η, ξ ) p 1 = ( ξ, η ) p 0 , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 14

In the Plane Fact The area of the (interpolating) optimal triangles in the plane is � 2 5 ϵ . p 2 = ( η, ξ ) p 1 = ( ξ, η ) p 0 , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 14

In the Plane Fact The area of the (interpolating) optimal triangles in the plane is � 2 5 ϵ . p 2 = ( η, ξ ) p 1 = ( ξ, η ) p 0 , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 14

In the Plane Fact The area of the (interpolating) optimal triangles in the plane is � 2 5 ϵ . p 2 = ( η, ξ ) p 1 = ( ξ, η ) p 0 , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 14

In the Plane Fact The area of the (interpolating) optimal triangles in the plane is � 2 5 ϵ . p 2 = ( η, ξ ) ◮ one-parameter family of area preserving triangles p 1 = ( ξ, η ) p 0 , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 14