Geometric Modeling with Conical Meshes and Developable Surfaces Siggraph 2006

1
Geometric Modeling with Conical Meshes and
Developable SurfacesSiggraph 2006
2
Authors

• Yang Liu Univ. of Hong Kong
• Helmut Pottmann TU Wien
• Johannes Wallner TU Wien
• Yong-Liang Yang Tsinghua Univ., Beijing
• Wenping Wang Univ. of Hong Kong

3
Outline

• Introduction
• PQ Meshes
• PQ Perturbation
• Subdivision and PQ meshes
• Conical Meshes
• Results

4
Introduction
5
PQ Meshes (I)

• Quad meshes with planar faces
• PQ strip
• 1. ai bi are parallel

b0
b1
bn
a1
an
a0
6
PQ Meshes (I)

• Quad meshes with planar faces
• PQ strip
• 2. ai bi pass through a fixed point s

s
bn
b0
an
b1
a0
a1
7
PQ Meshes (I)

• Quad meshes with planar faces
• PQ strip
• 3. PQ strip is a patch on the tangent surface on
  a polyline r1,rn

8
PQ Meshes (II)

• Assume interior mesh vertices have valence 4

9
PQ Perturbation (I)

• We use 4 terms to achieve a PQ mesh.
• 2 for planarity, 1 for fairness and 1 for
  closeness to original mesh.

10
PQ Perturbation - Planarity
3
1
2
4
11
PQ Perturbation - Planarity
ei, j1
ei1, j
ei, j
ei1, j1
12
PQ Perturbation - Fairness

• At boundary not all vertices required by the sum
  exist.

13
PQ Perturbation - Closeness

• yi, j is the closest point on mesh to vi, j

14
PQ Perturbation - Solve

• Define Lagragian function
• Minimizes w1ffair w2fclose
• Subject to the constraints cpq 0 and cdet 0

15
PQ Perturbation - Solve

• J for Jacobian matrix of constraints c(x)
• H for Hessian matrix of fPQ (x, ?)
• Update x ? x h from

16
For example
17
Subdivision and PQ Meshes
18
Subdivision and PQ Meshes

• Alternating application of PQ Perturbation and
  Subdivision.

19
Conical Meshes (I)

• Conical vertex if all 4 face planes meeting at v
  are tangent to a common cone of revolution.

20
Conical Meshes (II)

• We call a PQ mesh a conical mesh if all of its
  vertices of valence 4 are conical.
• Fact 1 A conical mesh have the property that
  offsetting all faces by a fixed distance leads
  again to a mesh with the same connectivity.

21
Conical Meshes (II)

• Fact 2 Successive discrete normals of a conical
  mesh along a row or column are co-planar.
• Fact 3 If a subdivision process, which preserves
  the conical property.

22
Conical Meshes (III)
w3
c
b
c
w4
b
d
d
a
w2
a
w1
23
Conical Meshes (IV)

• Add new constraint
• W1 W3 W2 W4 0

24
Conical Meshes (V)
25
Results
26
Results
27
Thanks

• _