SLIDE 1 (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
SLIDE 2 Motivation
◮ Smooth surface is locally
approximated by a quadratic patch.
◮ Euclidean motion transforms
the quadratic patch to graph
- f 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
SLIDE 3
Outline
Introduction Interpolating Approximation Non-interpolating Approximation
, Optimal Triangulation ➢ Introduction SFB DGD Workshop 3
SLIDE 4
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 D1, D2 ⊂ R2 and two graphs f : D1 → R and g: D2 → R then the vertical distance is distV (f, g) = max
(x,y)∈D1∩D2
|f(x, y) − g(x, y)|
, Optimal Triangulation ➢ Introduction SFB DGD Workshop 4
SLIDE 5
Properties of V-Distance Lemma
Let A, B ⊂ R3 be two sets with equal projection to the plane. Then distH (A, B) ≤ distV (A, B) v h
α α , Optimal Triangulation ➢ Introduction SFB DGD Workshop 5
SLIDE 6
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 Tp: R3 → R3 which satisfies the following:
◮ Tp(p) = ◮ Tp(S) = a quadratic graph ˜
S with a homogeneous polynomial of the form ˜ F(x, y) = ax2 + bxy + cy2 (∗)
◮ For all q, r ∈ R3 on a vertical line,
|q − r| = |Tp(q) − Tp(r)|.
◮ Tp(p) on the first two coordinates is a translation in R2.
, Optimal Triangulation ➢ Introduction SFB DGD Workshop 6
SLIDE 7 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 = (px, py, pz) and q = (qx, qy, qz) on a quadratic graph S, distV (pq, S) = 1
4
F(qx − px, qy − py)
F(x, y) is the homogeneous polynomial (∗).
◮ The max. vertical distance is attained at the midpoint.
p q
, Optimal Triangulation ➢ Introduction SFB DGD Workshop 7
SLIDE 8 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 p0, p1, p2 ∈ S of largest area such that distV (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
SLIDE 9 Maximize the Area of Planar Triangles
| x y | = 4 ϵ
p0
, Optimal Triangulation ➢ Interpolating Approximation SFB DGD Workshop 9
SLIDE 10 Maximize the Area of Planar Triangles
| x y | = 4 ϵ
p0 p1
, Optimal Triangulation ➢ Interpolating Approximation SFB DGD Workshop 9
SLIDE 11 Maximize the Area of Planar Triangles
| x y | = 4 ϵ | ( x − ξ ) y | = 4 ϵ
p0 p1
, Optimal Triangulation ➢ Interpolating Approximation SFB DGD Workshop 9
SLIDE 12 Maximize the Area of Planar Triangles
| x y | = 4 ϵ | ( x − ξ ) y | = 4 ϵ
p0 p1
p2,1 p2,2 p2,3 p2,4 p2,5 p2,6
, Optimal Triangulation ➢ Interpolating Approximation SFB DGD Workshop 9
SLIDE 13 Maximize the Area of Planar Triangles
| x y | = 4 ϵ | ( x − ξ ) y | = 4 ϵ
p0 p1
T4(ξ)
p2,1 p2,2 p2,3 p2,4 p2,5 p2,6
, Optimal Triangulation ➢ Interpolating Approximation SFB DGD Workshop 9
SLIDE 14
Optimize the Shape of Planar Triangles
Secondary criterion: Maximize the smallest angle x y
, Optimal Triangulation ➢ Interpolating Approximation SFB DGD Workshop 10
SLIDE 15
Optimize the Shape of Planar Triangles
Secondary criterion: Maximize the smallest angle x y
, Optimal Triangulation ➢ Interpolating Approximation SFB DGD Workshop 10
SLIDE 16 Triangulate the Saddle
Lift the planar triangulation to the surface
x y x y
, Optimal Triangulation ➢ Interpolating Approximation SFB DGD Workshop 11
SLIDE 17 Can We Do Better? What do we have?
Given an ϵ > 0 and a saddle surface S, we can find a family T
- f triangles which interpolate the surface and
◮ have maximum area, ◮ maintain distV (S, T) ≤ ϵ for all T ∈ T .
, Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 12
SLIDE 18 Can We Do Better? What do we have?
Given an ϵ > 0 and a saddle surface S, we can find a family T
- f triangles which interpolate the surface and
◮ have maximum area, ◮ maintain distV (S, T) ≤ ϵ for all T ∈ T .
◮ 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
SLIDE 19
Pseudo-Euclidean Transformations
◮ A λ-pseudo Euclidean map is
given by: (x, y) → (λx, 1
λy) ◮ Vertical distance is preserved. ◮ Area (projected) is preserved. ◮ Surface S = { z = xy } is
preserved.
, Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 13
SLIDE 20
Pseudo-Euclidean Transformations
◮ A λ-pseudo Euclidean map is
given by: (x, y) → (λx, 1
λy) ◮ Vertical distance is preserved. ◮ Area (projected) is preserved. ◮ Surface S = { z = xy } is
preserved.
p0 p1 p2 , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 13
SLIDE 21
Pseudo-Euclidean Transformations
◮ A λ-pseudo Euclidean map is
given by: (x, y) → (λx, 1
λy) ◮ Vertical distance is preserved. ◮ Area (projected) is preserved. ◮ Surface S = { z = xy } is
preserved.
p0 p1 p2 , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 13
SLIDE 22
Pseudo-Euclidean Transformations
◮ A λ-pseudo Euclidean map is
given by: (x, y) → (λx, 1
λy) ◮ Vertical distance is preserved. ◮ Area (projected) is preserved. ◮ Surface S = { z = xy } is
preserved.
p0 p1 p2 , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 13
SLIDE 23
Pseudo-Euclidean Transformations
◮ A λ-pseudo Euclidean map is
given by: (x, y) → (λx, 1
λy) ◮ Vertical distance is preserved. ◮ Area (projected) is preserved. ◮ Surface S = { z = xy } is
preserved.
p0 p1 p2 , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 13
SLIDE 24
Pseudo-Euclidean Transformations
◮ A λ-pseudo Euclidean map is
given by: (x, y) → (λx, 1
λy) ◮ Vertical distance is preserved. ◮ Area (projected) is preserved. ◮ Surface S = { z = xy } is
preserved.
p0 p1 p2 , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 13
SLIDE 25 In the Plane Fact
The area of the (interpolating) optimal triangles in the plane is 2
below S above S
, Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 14
SLIDE 26 In the Plane Fact
The area of the (interpolating) optimal triangles in the plane is 2
p0 p1 = (ξ, η) p2 = (η, ξ) , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 14
SLIDE 27 In the Plane Fact
The area of the (interpolating) optimal triangles in the plane is 2
p0 p1 = (ξ, η) p2 = (η, ξ) , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 14
SLIDE 28 In the Plane Fact
The area of the (interpolating) optimal triangles in the plane is 2
p0 p1 = (ξ, η) p2 = (η, ξ) , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 14
SLIDE 29 In the Plane Fact
The area of the (interpolating) optimal triangles in the plane is 2
p0 p1 = (ξ, η) p2 = (η, ξ) , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 14
SLIDE 30 In the Plane Fact
The area of the (interpolating) optimal triangles in the plane is 2
p0 p1 = (ξ, η) p2 = (η, ξ) , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 14
SLIDE 31 In the Plane Fact
The area of the (interpolating) optimal triangles in the plane is 2
p0 p1 = (ξ, η) p2 = (η, ξ) , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 14
SLIDE 32 In the Plane Fact
The area of the (interpolating) optimal triangles in the plane is 2
◮ one-parameter family
triangles
p0 p1 = (ξ, η) p2 = (η, ξ) , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 14
SLIDE 33 In the Plane Fact
The area of the (interpolating) optimal triangles in the plane is 2
◮ one-parameter family
triangles
◮ How should they be
lifted?
p0 p1 = (ξ, η) p2 = (η, ξ) , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 14
SLIDE 34
Vertical Perturbed Lifting
◮ Lift the triangle vertically such
that the distance to S is minimized.
p0 p1 = (ξ, η) p2 = (η, ξ) , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 15
SLIDE 35
Vertical Perturbed Lifting
◮ Lift the triangle vertically such
that the distance to S is minimized.
◮ Lift vertices off the surface
by α: Sα = (x, y, z) : z = xy + α
p0 p1 = (ξ, η) p2 = (η, ξ) , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 15
SLIDE 36
Vertical Perturbed Lifting
◮ Lift the triangle vertically such
that the distance to S is minimized.
◮ Lift vertices off the surface
by α: Sα = (x, y, z) : z = xy + α
◮ Vertical distance is attained at
midpoints.
p0 p1 = (ξ, η) p2 = (η, ξ) , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 15
SLIDE 37
Vertical Perturbed Lifting (Cont.)
◮ Vertical distances from edges
to S are ξη 4 + α > 0 1 4 (ξ − η)2 − α > 0 and have to be equal.
p0 p1 = (ξ, η) p2 = (η, ξ) , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 16
SLIDE 38
Vertical Perturbed Lifting (Cont.)
◮ Vertical distances from edges
to S are ξη 4 + α > 0 1 4 (ξ − η)2 − α > 0 and have to be equal.
◮ α =
1 8 (ξ2 − 3ξη + η2)
p0 p1 = (ξ, η) p2 = (η, ξ) , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 16
SLIDE 39 Vertical Perturbed Lifting (Cont.)
◮ The vertical distance is
8 (ξ2 − ξη + η2)
p1 = (ξ, η) p2 = (η, ξ) , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 17
SLIDE 40 Vertical Perturbed Lifting (Cont.)
◮ The vertical distance is
8 (ξ2 − ξη + η2)
- ◮ Minimum is attained for
ξ0 =
2 +
p0 p1 = (ξ, η) p2 = (η, ξ) , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 17
SLIDE 41 Vertical Perturbed Lifting (Cont.)
◮ The vertical distance is
8 (ξ2 − ξη + η2)
- ◮ Minimum is attained for
ξ0 =
2 +
◮ and the vertical distance is
4 ϵ ≈ 0.968246ϵ
p0 p1 = (ξ, η) p2 = (η, ξ) , Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 17
SLIDE 42
Picture in Space
, Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 18
SLIDE 43 The Planar Super-Optimal Triangle
◮ Pseudo-euclidean motions
give a one-parameter family
, Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 19
SLIDE 44 The Planar Super-Optimal Triangle
◮ Pseudo-euclidean motions
give a one-parameter family
◮ Note the tangency property
, Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 19
SLIDE 45 The Planar Super-Optimal Triangle
◮ Pseudo-euclidean motions
give a one-parameter family
◮ Note the tangency property
OPEN: Lift vertices by different amounts?
, Optimal Triangulation ➢ Non-interpolating Approximation SFB DGD Workshop 19
