Henrik Zimmer Computer Graphics Group
Variational Tangent Plane Intersection for Planar Polygonal Meshing
Henrik Zimmer, Marcel Campen, Ralf Herkrath, Leif Kobbelt
1
Henrik Zimmer Computer Graphics Group
Motivation
2
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
3
Geometry of multi-layer freeform structures for architecture. Pottmann, Liu, Wallner, Bobenko, Wang. 2007
Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006
Discrete Surfaces for Architectural Design. Pottmann, Brell-Cokcan, Wallner. 2006
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
4
Geometry of multi-layer freeform structures for architecture. Pottmann, Liu, Wallner, Bobenko, Wang. 2007
Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006
Discrete Surfaces for Architectural Design. Pottmann, Brell-Cokcan, Wallner. 2006
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
5
Geometry of multi-layer freeform structures for architecture. Pottmann, Liu, Wallner, Bobenko, Wang. 2007
Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006
Discrete Surfaces for Architectural Design. Pottmann, Brell-Cokcan, Wallner. 2006
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
6
Geometry of multi-layer freeform structures for architecture. Pottmann, Liu, Wallner, Bobenko, Wang. 2007
Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006
Discrete Surfaces for Architectural Design. Pottmann, Brell-Cokcan, Wallner. 2006
Supporting + Covering Layer
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
7
Node/edge simplicity
Geometry of multi-layer freeform structures for architecture. Pottmann, Liu, Wallner, Bobenko, Wang. 2007
Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006
Discrete Surfaces for Architectural Design. Pottmann, Brell-Cokcan, Wallner. 2006
Supporting + Covering Layer
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
8
Planar panels Node/edge simplicity
Geometry of multi-layer freeform structures for architecture. Pottmann, Liu, Wallner, Bobenko, Wang. 2007
Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006
Discrete Surfaces for Architectural Design. Pottmann, Brell-Cokcan, Wallner. 2006
Supporting + Covering Layer
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
9
Planar panels Node/edge simplicity
Geometry of multi-layer freeform structures for architecture. Pottmann, Liu, Wallner, Bobenko, Wang. 2007
Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006
Discrete Surfaces for Architectural Design. Pottmann, Brell-Cokcan, Wallner. 2006
Supporting + Covering Layer
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
• Multi-Layer Dual Structures
10
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
• Multi-Layer Dual Structures
11
Primal Layer
Henrik Zimmer Computer Graphics Group
Primal Layer
Dual Layer
Motivation
• Multi-Layer Support Structures
• Multi-Layer Dual Structures
12
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
• Multi-Layer Dual Structures
13
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
• Multi-Layer Dual Structures
• Tangent Plane Intersection (TPI)
14
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
• Multi-Layer Dual Structures
• Tangent Plane Intersection (TPI)
15
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
• Multi-Layer Dual Structures
• Tangent Plane Intersection (TPI)
• Variational TPl
16
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
• Multi-Layer Dual Structures
• Tangent Plane Intersection (TPI)
• Variational TPl
– Polygon Mesh Planarization
17
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
• Multi-Layer Dual Structures
• Tangent Plane Intersection (TPI)
• Variational TPl
– Dual Support Structures
18
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
• Multi-Layer Dual Structures
• Tangent Plane Intersection (TPI)
• Variational TPl
– Variational Shape Approximation
19
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
• Multi-Layer Dual Structures
• Tangent Plane Intersection (TPI)
• Variational TPl
20
Planarization, Dual Support Structures, Shape Approximation
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
• Multi-Layer Dual Structures
• Tangent Plane Intersection (TPI)
• Variational TPl
21
Planarization, Dual Support Structures, Shape Approximation
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
• Multi-Layer Dual Structures
• Tangent Plane Intersection (TPI)
• Variational TPl
22
Planarization, Dual Support Structures, Shape Approximation
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
• Multi-Layer Dual Structures
• Tangent Plane Intersection (TPI)
• Variational TPl
23
Planarization, Dual Support Structures, Shape Approximation
Henrik Zimmer Computer Graphics Group
Motivation
• Multi-Layer Support Structures
• Multi-Layer Dual Structures
• Tangent Plane Intersection (TPI)
• Variational TPl
24
Planarization, Dual Support Structures, Shape Approximation
Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• 3 planes necessary for unique intersection point
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Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• 3 planes necessary for unique intersection point
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∩ =
∩ =
∩ =
Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• 3 planes necessary for unique intersection point
27
∩ =
∩ =
∩ =
[Planar Hexagonal Meshes by Tangent Plane Intersection, Troche 2008]
Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• 3 planes necessary for unique intersection point
28
[Planar Hexagonal Meshes by Tangent Plane Intersection, Troche 2008]
𝒏0
𝒗0
𝒏2
𝒗2
𝒏1
𝒗1 𝒏0𝑇
𝒏1𝑇
𝒏2𝑇
𝒙 =
𝒏0𝑇𝒗0𝒏1𝑇𝒗1𝒏2𝑇𝒗2
𝑁𝒙 = 𝒃
Intersection point 𝒙 obtained by inversion
𝒙 = 𝑁−1𝒃
Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• 3 planes necessary for unique intersection point
– Positive Curvature: OK
29
[Planar Hexagonal Meshes by Tangent Plane Intersection, Troche 2008]
Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• 3 planes necessary for unique intersection point
– Low Gaussian Curvature: UNSTABLE
30
[Planar Hexagonal Meshes by Tangent Plane Intersection, Troche 2008]
Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• 3 planes necessary for unique intersection point
– Low Gaussian Curvature: UNSTABLE
31
[Planar Hexagonal Meshes by Tangent Plane Intersection, Troche 2008]
Variational Tangent Plane Intersection
Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• 3 planes necessary for unique intersection point
– Intersections are predetermined: No design DoFs
32
[Planar Hexagonal Meshes by Tangent Plane Intersection, Troche 2008]
Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• 3 planes necessary for unique intersection point
– Intersections are predetermined: No design DoFs
33
[Planar Hexagonal Meshes by Tangent Plane Intersection, Troche 2008]
Variational Tangent Plane Intersection
Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• Problem 1: Instability, co-planar tangent planes
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Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• Problem 1: Instability, co-planar tangent planes
35
Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• Problem 1: Instability, co-planar tangent planes
36
Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• Problem 1: Instability, co-planar tangent planes
– Intersection point 𝒙 = 𝑁−1𝒃 is not well-defined
37
Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• Problem 1: Instability, co-planar tangent planes
– Intersection point 𝒙 = 𝑁−1𝒃 is not well-defined
– However, 𝑁𝒙 = 𝒃 still holds for all points
38
Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• Problem 1: Instability, co-planar tangent planes
– Intersection point 𝒙 = 𝑁−1𝒃 is not well-defined
– However, 𝑁𝒙 = 𝒃 still holds for all points
from all points we would like to choose our favorite
39
Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• Problem 2: bad (predetermined) intersections
– Intersection is well-defined but position unwanted
40
Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• Problem 2: bad (predetermined) intersections
– Intersection is well-defined but position unwanted
– Could obtain more degrees of freedom by:
• rotating tangent planes
41
Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• Problem 2: bad (predetermined) intersections
– Intersection is well-defined but position unwanted
– Could obtain more degrees of freedom by:
• rotating tangent planes
42
Henrik Zimmer Computer Graphics Group
Tangent Plane Intersection
• Problem 2: bad (predetermined) intersections
– Intersection is well-defined but position unwanted
– Could obtain more degrees of freedom by:
• rotating tangent planes
• offsetting tangent planes
43
Henrik Zimmer Computer Graphics Group
Variational Tangent Plane Intersection
44
Henrik Zimmer Computer Graphics Group
Variational Tangent Plane Intersection
• Let 𝑀 = (𝑉, 𝐹) be a triangle mesh
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Henrik Zimmer Computer Graphics Group
Variational Tangent Plane Intersection
• Let 𝑀 = (𝑉, 𝐹) be a triangle mesh
• Formulate TPI as a constrained optimization:
where 𝒙𝑓 are the unknown intersection points
46
minimize 𝐸 s.t. ∀𝑓 ∈ 𝐹 𝐶int: 𝑁𝑓𝒙𝑓 = 𝒃𝑓
Henrik Zimmer Computer Graphics Group
Variational Tangent Plane Intersection
• Let 𝑀 = (𝑉, 𝐹) be a triangle mesh
• Formulate TPI as a constrained optimization:
where 𝒙𝑓 are the unknown intersection points
• In non-degenerate configurations the solution is equivalent to the explicit TPI approach
47
minimize 𝐸 s.t. ∀𝑓 ∈ 𝐹 𝐶int: 𝑁𝑓𝒙𝑓 = 𝒃𝑓
Henrik Zimmer Computer Graphics Group
Variational Tangent Plane Intersection
49
s.t. 𝐸 ∀𝑓 ∈ 𝐹 𝐶int: 𝑁𝑓𝒙𝑓 = 𝒃𝑓
Henrik Zimmer Computer Graphics Group
Variational Tangent Plane Intersection
• Solution to Problem 1: instability
– Specify energy with preferred intersection points 𝒑𝑓
50
s.t. 𝐸 ≔ 𝒙𝑓 − 𝒑𝑓2
𝑓∈𝐹
∀𝑓 ∈ 𝐹 𝐶int: 𝑁𝑓𝒙𝑓 = 𝒃𝑓
Henrik Zimmer Computer Graphics Group
Variational Tangent Plane Intersection
• Solution to Problem 1: instability
– Specify energy with preferred intersection points 𝒑𝑓
• Solution to Problem 2: “bad” intersection points
– Introduce variable offsets ℎ𝑣
51
s.t. 𝐸 ≔ 𝒙𝑓 − 𝒑𝑓2
𝑓∈𝐹
∀𝑓 ∈ 𝐹 𝐶int: 𝑁𝑓𝒙𝑓 = 𝒃𝑓 − 𝒉𝑓
Henrik Zimmer Computer Graphics Group
Variational Tangent Plane Intersection
• Solution to Problem 1: instability
– Specify energy with preferred intersection points 𝒑𝑓
• Solution to Problem 2: “bad” intersection points
– Introduce variable offsets ℎ𝑣 and normals 𝒏𝑣
52
s.t. 𝐸 ≔ 𝒙𝑓 − 𝒑𝑓2
𝑓∈𝐹
∀𝑣 ∈ 𝑉
∀𝑓 ∈ 𝐹 𝐶int:
𝐶norm: 𝒏𝑣2 = 1
𝑁𝑓𝒙𝑓 = 𝒃𝑓 − 𝒉𝑓
Henrik Zimmer Computer Graphics Group
Variational Tangent Plane Intersection
• Solution to Problem 1: instability
– Specify energy with preferred intersection points 𝒑𝑓
• Solution to Problem 2: “bad” intersection points
– Introduce variable offsets ℎ𝑣 and normals 𝒏𝑣
• So far VTPI is defined for triangle meshes …
53
s.t. 𝐸 ≔ 𝒙𝑓 − 𝒑𝑓2
𝑓∈𝐹
∀𝑣 ∈ 𝑉
∀𝑓 ∈ 𝐹 𝐶int:
𝐶norm: 𝒏𝑣2 = 1
𝑁𝑓𝒙𝑓 = 𝒃𝑓 − 𝒉𝑓
Henrik Zimmer Computer Graphics Group
Multiple Tangent Plane Intersection
54
Henrik Zimmer Computer Graphics Group
𝒏0𝑇
𝒏1𝑇
𝒏2𝑇
𝒙 =
𝒏0𝑇𝒗0𝒏1𝑇𝒗1𝒏2𝑇𝒗2
𝑁𝒙 = 𝒃
Multiple Tangent Plane Intersection
• 3 planes necessary for unique intersection point
55
𝒏0
𝒗0
𝒏2
𝒗2
𝒏1
𝒗1
Henrik Zimmer Computer Graphics Group
Multiple Tangent Plane Intersection
• 3 planes necessary for unique intersection point
56
𝒏0
𝒗0
𝒏2
𝒗2
𝒏1
𝒗1
𝒏3
𝒗3
𝒏0𝑇
𝒏1𝑇
𝒏2𝑇
𝒙 =
𝒏0𝑇𝒗0𝒏1𝑇𝒗1𝒏2𝑇𝒗2
𝑁𝒙 = 𝒃
Henrik Zimmer Computer Graphics Group
Multiple Tangent Plane Intersection
• 3 planes necessary for unique intersection point
• No longer limited to triangle meshes
57
𝒏0
𝒗0
𝒏2
𝒗2
𝒏1
𝒗1
𝒏3
𝒗3
𝒏0𝑇
𝒏1𝑇
𝒏2𝑇
𝒏3𝑇
𝒙 =
𝒏0𝑇𝒗0𝒏1𝑇𝒗1𝒏2𝑇𝒗2𝒏3𝑇𝒗3
𝑁𝒙 = 𝒃
Henrik Zimmer Computer Graphics Group
𝒏0𝑇
𝒏1𝑇
𝒏2𝑇
𝒏3𝑇
⋮
𝒙 =
𝒏0𝑇𝒗0𝒏1𝑇𝒗1𝒏2𝑇𝒗2𝒏3𝑇𝒗3⋮
𝑁𝒙 = 𝒃
Multiple Tangent Plane Intersection
• 3 planes necessary for unique intersection point
• Works on general polygon meshes
58
𝒏0
𝒗0
𝒏2
𝒗2
𝒏1
𝒗1
𝒏3
𝒗3
Henrik Zimmer Computer Graphics Group
VTPI for Polygon Mesh Planarization
59
Henrik Zimmer Computer Graphics Group
VTPI for Polygon Mesh Planarization
• 𝑀 ↦ planar dual(𝑀)
60
Henrik Zimmer Computer Graphics Group
VTPI for Polygon Mesh Planarization
• 𝑀 ↦ planar dual(𝑀)
• dual(𝑀) ↦ planar 𝑀
61
Henrik Zimmer Computer Graphics Group
VTPI for Polygon Mesh Planarization
• 𝑀 ↦ planar dual(𝑀)
• dual(𝑀) ↦ planar 𝑀
62
F E R T I L I T Y
Henrik Zimmer Computer Graphics Group
VTPI for Polygon Mesh Planarization
• 𝑀 ↦ planar dual(𝑀)
• dual(𝑀) ↦ planar 𝑀
63
[Variational Tangent Plane Intersection] F E R T I L I T Y
Henrik Zimmer Computer Graphics Group
VTPI for Polygon Mesh Planarization
• 𝑀 ↦ planar dual(𝑀)
• dual(𝑀) ↦ planar 𝑀
64
[Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011]
[Variational Tangent Plane Intersection] F E R T I L I T Y
Henrik Zimmer Computer Graphics Group
VTPI for Polygon Mesh Planarization
• 𝑀 ↦ planar dual(𝑀)
• dual(𝑀) ↦ planar 𝑀
• Consider quad strips undergoing twists
65
[Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011]
[Variational Tangent Plane Intersection] F E R T I L I T Y
Henrik Zimmer Computer Graphics Group
VTPI for Polygon Mesh Planarization
• 𝑀 ↦ planar dual(𝑀)
• dual(𝑀) ↦ planar 𝑀
• Consider quad strips undergoing twists
66
[Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011]
[Variational Tangent Plane Intersection] F E R T I L I T Y
Henrik Zimmer Computer Graphics Group
VTPI for Polygon Mesh Planarization
• 𝑀 ↦ planar dual(𝑀)
• dual(𝑀) ↦ planar 𝑀
67
[Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011]
[Variational Tangent Plane Intersection] F E L I N E
Henrik Zimmer Computer Graphics Group
VTPI for Polygon Mesh Planarization
• 𝑀 ↦ planar dual(𝑀)
• dual(𝑀) ↦ planar 𝑀
• Consider quads not aligned to princ. curvatures
68
[Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011]
[Variational Tangent Plane Intersection] F E L I N E
Henrik Zimmer Computer Graphics Group
VTPI for Polygon Mesh Planarization
• 𝑀 ↦ planar dual(𝑀)
• dual(𝑀) ↦ planar 𝑀
• Consider quads not aligned to princ. curvatures
69
[Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011]
[Variational Tangent Plane Intersection] F E L I N E
Henrik Zimmer Computer Graphics Group
VTPI for Polygon Mesh Planarization
• Comparison to other planarization techniques
– Optimization (perturbation)-based methods
– Planarizing flow
70
Method
VTPI Opt. Flow
Pro
pe
rty Parameters - - +
Extensions + + -
Normals + - -
[Hexagonal Meshes with Planar Faces. Wang, Liu, Yan, Chan, Ling, Sun. 2008]
[Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011]
[Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006]
Henrik Zimmer Computer Graphics Group
VTPI for Polygon Mesh Planarization
• Comparison to other planarization techniques
– Optimization (perturbation)-based methods
– Planarizing flow
71
Method
VTPI Opt. Flow
Pro
pe
rty Parameters - - +
Extensions + + -
Normals + - -
[Hexagonal Meshes with Planar Faces. Wang, Liu, Yan, Chan, Ling, Sun. 2008]
[Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011]
[Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006]
Henrik Zimmer Computer Graphics Group
VTPI for Polygon Mesh Planarization
• Comparison to other planarization techniques
– Optimization (perturbation)-based methods
– Planarizing flow
72
Method
VTPI Opt. Flow
Pro
pe
rty Parameters - - +
Extensions + + -
Normals + - -
[Hexagonal Meshes with Planar Faces. Wang, Liu, Yan, Chan, Ling, Sun. 2008]
[Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011]
[Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006]
Henrik Zimmer Computer Graphics Group
VTPI for Polygon Mesh Planarization
• Comparison to other planarization techniques
– Optimization (perturbation)-based methods
– Planarizing flow
73
Method
VTPI Opt. Flow
Pro
pe
rty Parameters - - +
Extensions + + -
Normals + - -
[Hexagonal Meshes with Planar Faces. Wang, Liu, Yan, Chan, Ling, Sun. 2008]
[Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011]
[Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006]
Henrik Zimmer Computer Graphics Group
VTPI for Polygon Mesh Planarization
• Comparison to other planarization techniques
– Optimization (perturbation)-based methods
– Planarizing flow
• use normals → intersection-free dual structures
74
Method
VTPI Opt. Flow
Pro
pe
rty Parameters - - +
Extensions + + -
Normals + - -
[Hexagonal Meshes with Planar Faces. Wang, Liu, Yan, Chan, Ling, Sun. 2008]
[Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011]
[Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006]
Henrik Zimmer Computer Graphics Group
VTPI for Multi-Layer Dual Structures
75
Input Dual TPI
VTPI VTPI + preferred offsets
VTPI + preferred offsets + intersection hard-constraints
Henrik Zimmer Computer Graphics Group
Avoiding (local) Intersections
76
Problem Solution (constraints)
Henrik Zimmer Computer Graphics Group
Avoiding (local) Intersections
77
Problem Solution (constraints)
Vert
ex
Inte
rse
ction
dual
primal
Henrik Zimmer Computer Graphics Group
Avoiding (local) Intersections
78
Problem Solution (constraints)
Vert
ex
Inte
rse
ction
dual
primal
ℎ>0
Henrik Zimmer Computer Graphics Group
Avoiding (local) Intersections
79
Problem Solution (constraints)
Vert
ex
Inte
rse
ction
F
ace
In
ters
ection
dual
primal
ℎ>0
Henrik Zimmer Computer Graphics Group
Avoiding (local) Intersections
80
Problem Solution (constraints)
Vert
ex
Inte
rse
ction
F
ace
In
ters
ection
dual
primal
ℎ>0
𝒎𝑇 𝒙−𝒗 >0
𝒎 𝒗
𝒙
Henrik Zimmer Computer Graphics Group
Avoiding (local) Intersections
81
Problem Solution (constraints)
Vert
ex
Inte
rse
ction
F
ace
In
ters
ection
E
dge
In
ters
ection
dual
primal
ℎ>0
𝒎𝑇 𝒙−𝒗 >0
𝒎 𝒗
𝒙
Henrik Zimmer Computer Graphics Group
Avoiding (local) Intersections
82
Problem Solution (constraints)
Vert
ex
Inte
rse
ction
F
ace
In
ters
ection
E
dge
In
ters
ection
dual
primal
ℎ>0
𝒎𝑇 𝒙−𝒗 >0
𝒎 𝒗
𝒙
𝒒𝑇𝒆<0 𝒆 𝒒
Henrik Zimmer Computer Graphics Group
Results of Multi-Layer Dual Structures
83
Henrik Zimmer Computer Graphics Group
Results of Multi-Layer Dual Structures
84
Co
urt
esy
of
ww
w.e
volu
te.a
t
T R A I N S TA T I O N
Henrik Zimmer Computer Graphics Group
Results of Multi-Layer Dual Structures
85
A L P I N E H U T
Side view
Back view
Henrik Zimmer Computer Graphics Group
Conclusion
• What is VTPI?
86
Henrik Zimmer Computer Graphics Group
Conclusion
• What is VTPI?
– A variational formulation of tangent plane intersection
• Guided intersections of several planes
• Useful for geometric problems (e.g. mesh planarization)
• Solved by global optimization (freely available solvers)
87
Henrik Zimmer Computer Graphics Group
Conclusion
• What is VTPI?
– A variational formulation of tangent plane intersection
• Guided intersections of several planes
• Useful for geometric problems (e.g. mesh planarization)
• Solved by global optimization (freely available solvers)
• What is VTPI not?
– A “fix” to topological issues involved in planar meshing
• Degeneracies will occur where necessary, e.g. concave (or degenerate) hexagons in hyperbolic surface regions
• Energies can sometimes be used to shift such effects …
88
Henrik Zimmer Computer Graphics Group
Limitations & Discussion
• Output depends on input tessellation and energy
89
Henrik Zimmer Computer Graphics Group
Limitations & Discussion
• Output depends on input tessellation and energy
– Different tessellations, same topology
90
Henrik Zimmer Computer Graphics Group
Limitations & Discussion
• Output depends on input tessellation and energy
– Different tessellations, same topology, same functional
91
Henrik Zimmer Computer Graphics Group
Limitations & Discussion
• Output depends on input tessellation and energy
– Different tessellations, same topology, same functional
92
Henrik Zimmer Computer Graphics Group
Limitations & Discussion
• Output depends on input tessellation and energy
– energies can partly shift some effects on the mesh
93
Henrik Zimmer Computer Graphics Group
Limitations & Discussion
• Output depends on input tessellation and energy
– Same tessellation, same topology, different functional
94
Normal Smoothness Element Fairing
Henrik Zimmer Computer Graphics Group
Limitations & Discussion
• Output depends on input tessellation and energy
– Same tessellation, same topology, different functional
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Normal Smoothness Element Fairing
Henrik Zimmer Computer Graphics Group
The End
Thank you for your attention!
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