slides - computer graphics and multimedia

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

25

Henrik Zimmer Computer Graphics Group

Tangent Plane Intersection

• 3 planes necessary for unique intersection point

26

∩ =

∩ =

∩ =

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

34

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

45

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

95

Normal Smoothness Element Fairing

Henrik Zimmer Computer Graphics Group

The End

Thank you for your attention!

96