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


Motivation

2


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


Motivation


4





Motivation


5





Motivation


6




Supporting + Covering Layer


Motivation


7

Node/edge simplicity






Motivation


8

Planar panels Node/edge simplicity






Motivation


9

Planar panels Node/edge simplicity






Motivation


• Multi-Layer Dual Structures

10


Motivation



11

Primal Layer


Primal Layer

Dual Layer

Motivation



12


Motivation



13


Motivation



• Tangent Plane Intersection (TPI)

14


Motivation




15


Motivation




• Variational TPl

16


Motivation




• Variational TPl

– Polygon Mesh Planarization

17


Motivation




• Variational TPl

– Dual Support Structures

18


Motivation




• Variational TPl

– Variational Shape Approximation

19


Motivation




• Variational TPl

20

Planarization, Dual Support Structures, Shape Approximation


Motivation




• Variational TPl

21



Motivation




• Variational TPl

22



Motivation




• Variational TPl

23



Motivation




• Variational TPl

24



Tangent Plane Intersection

• 3 planes necessary for unique intersection point

25




26

∩ =

∩ =

∩ =




27

∩ =

∩ =

∩ =

[Planar Hexagonal Meshes by Tangent Plane Intersection, Troche 2008]




28


𝒏0

𝒗0

𝒏2

𝒗2

𝒏1

𝒗1 𝒏0𝑇

𝒏1𝑇

𝒏2𝑇

𝒙 =

𝒏0𝑇𝒗0𝒏1𝑇𝒗1𝒏2𝑇𝒗2

𝑁𝒙 = 𝒃

Intersection point 𝒙 obtained by inversion

𝒙 = 𝑁−1𝒃




– Positive Curvature: OK

29





– Low Gaussian Curvature: UNSTABLE

30





– Low Gaussian Curvature: UNSTABLE

31


Variational Tangent Plane Intersection




– Intersections are predetermined: No design DoFs

32





– Intersections are predetermined: No design DoFs

33





• Problem 1: Instability, co-planar tangent planes

34




35




36




– Intersection point 𝒙 = 𝑁−1𝒃 is not well-defined

37





– However, 𝑁𝒙 = 𝒃 still holds for all points

38





– However, 𝑁𝒙 = 𝒃 still holds for all points

from all points we would like to choose our favorite

39



• Problem 2: bad (predetermined) intersections

– Intersection is well-defined but position unwanted

40





– Could obtain more degrees of freedom by:

• rotating tangent planes

41







42







• offsetting tangent planes

43



44



• Let 𝑀 = (𝑉, 𝐹) be a triangle mesh

45




• Formulate TPI as a constrained optimization:

where 𝒙𝑓 are the unknown intersection points

46

minimize 𝐸 s.t. ∀𝑓 ∈ 𝐹 𝐶int: 𝑁𝑓𝒙𝑓 = 𝒃𝑓




• 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



49

s.t. 𝐸 ∀𝑓 ∈ 𝐹 𝐶int: 𝑁𝑓𝒙𝑓 = 𝒃𝑓



• 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





• Solution to Problem 2: “bad” intersection points

– Introduce variable offsets ℎ𝑣

51


𝑓∈𝐹

∀𝑓 ∈ 𝐹 𝐶int: 𝑁𝑓𝒙𝑓 = 𝒃𝑓 − 𝒉𝑓






– Introduce variable offsets ℎ𝑣 and normals 𝒏𝑣

52


𝑓∈𝐹

∀𝑣 ∈ 𝑉

∀𝑓 ∈ 𝐹 𝐶int:

𝐶norm: 𝒏𝑣2 = 1

𝑁𝑓𝒙𝑓 = 𝒃𝑓 − 𝒉𝑓






– Introduce variable offsets ℎ𝑣 and normals 𝒏𝑣

• So far VTPI is defined for triangle meshes …

53


𝑓∈𝐹

∀𝑣 ∈ 𝑉

∀𝑓 ∈ 𝐹 𝐶int:

𝐶norm: 𝒏𝑣2 = 1

𝑁𝑓𝒙𝑓 = 𝒃𝑓 − 𝒉𝑓


Multiple Tangent Plane Intersection

54


𝒏0𝑇

𝒏1𝑇

𝒏2𝑇

𝒙 =


𝑁𝒙 = 𝒃



55

𝒏0

𝒗0

𝒏2

𝒗2

𝒏1

𝒗1




56

𝒏0

𝒗0

𝒏2

𝒗2

𝒏1

𝒗1

𝒏3

𝒗3

𝒏0𝑇

𝒏1𝑇

𝒏2𝑇

𝒙 =


𝑁𝒙 = 𝒃




• No longer limited to triangle meshes

57

𝒏0

𝒗0

𝒏2

𝒗2

𝒏1

𝒗1

𝒏3

𝒗3

𝒏0𝑇

𝒏1𝑇

𝒏2𝑇

𝒏3𝑇

𝒙 =

𝒏0𝑇𝒗0𝒏1𝑇𝒗1𝒏2𝑇𝒗2𝒏3𝑇𝒗3

𝑁𝒙 = 𝒃


𝒏0𝑇

𝒏1𝑇

𝒏2𝑇

𝒏3𝑇

⋮

𝒙 =

𝒏0𝑇𝒗0𝒏1𝑇𝒗1𝒏2𝑇𝒗2𝒏3𝑇𝒗3⋮

𝑁𝒙 = 𝒃



• Works on general polygon meshes

58

𝒏0

𝒗0

𝒏2

𝒗2

𝒏1

𝒗1

𝒏3

𝒗3


VTPI for Polygon Mesh Planarization

59



• 𝑀 ↦ planar dual(𝑀)

60




• dual(𝑀) ↦ planar 𝑀

61





62

F E R T I L I T Y





63

[Variational Tangent Plane Intersection] F E R T I L I T Y





64

[Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011]






• Consider quad strips undergoing twists

65







• Consider quad strips undergoing twists

66







67


[Variational Tangent Plane Intersection] F E L I N E





• Consider quads not aligned to princ. curvatures

68







• Consider quads not aligned to princ. curvatures

69





• 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]


[Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006]






71

Method

VTPI Opt. Flow

Pro

pe


Extensions + + -

Normals + - -









72

Method

VTPI Opt. Flow

Pro

pe


Extensions + + -

Normals + - -









73

Method

VTPI Opt. Flow

Pro

pe


Extensions + + -

Normals + - -









• use normals → intersection-free dual structures

74

Method

VTPI Opt. Flow

Pro

pe


Extensions + + -

Normals + - -





VTPI for Multi-Layer Dual Structures

75

Input Dual TPI

VTPI VTPI + preferred offsets

VTPI + preferred offsets + intersection hard-constraints


Avoiding (local) Intersections

76

Problem Solution (constraints)



77


Vert

ex

Inte

rse

ction

dual

primal



78


Vert

ex

Inte

rse

ction

dual

primal

ℎ>0



79


Vert

ex

Inte

rse

ction

F

ace

In

ters

ection

dual

primal

ℎ>0



80


Vert

ex

Inte

rse

ction

F

ace

In

ters

ection

dual

primal

ℎ>0

𝒎𝑇 𝒙−𝒗 >0

𝒎 𝒗

𝒙



81


Vert

ex

Inte

rse

ction

F

ace

In

ters

ection

E

dge

In

ters

ection

dual

primal

ℎ>0


𝒎 𝒗

𝒙



82


Vert

ex

Inte

rse

ction

F

ace

In

ters

ection

E

dge

In

ters

ection

dual

primal

ℎ>0


𝒎 𝒗

𝒙

𝒒𝑇𝒆<0 𝒆 𝒒


Results of Multi-Layer Dual Structures

83



84

Co

urt

esy

of

ww

w.e

volu

te.a

t

T R A I N S TA T I O N



85

A L P I N E H U T

Side view

Back view


Conclusion

• What is VTPI?

86


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


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


Limitations & Discussion

• Output depends on input tessellation and energy

89




– Different tessellations, same topology

90




– Different tessellations, same topology, same functional

91




– Different tessellations, same topology, same functional

92




– energies can partly shift some effects on the mesh

93




– Same tessellation, same topology, different functional

94

Normal Smoothness Element Fairing




– Same tessellation, same topology, different functional

95

Normal Smoothness Element Fairing


The End

Thank you for your attention!

96

slides - computer graphics and multimedia

Documents