Tao Ju A general construction of coordinates Slide 1

A general, geometric construction of coordinates in a

convex simplicial polytope

A general, geometric construction of coordinates in a

convex simplicial polytope

Tao Ju

Washington University in St. Louis

Tao Ju A general construction of coordinates Slide 2

CoordinatesCoordinates

• Homogeneous coordinates

– Given points

– Express a new point as affine combination of

– are called homogeneous coordinates

– Barycentric if all

},,,{ 1 ivvv

x v

1, iii bwherevbx 1, iii bwherevbx

0ibib

iv

x

Tao Ju A general construction of coordinates Slide 3

ApplicationsApplications

• Boundary interpolation

– Color/Texture interpolation

• Mapping

– Shell texture

– Image/Shape deformation

ii fbxf )( ii fbxf )(

iivbx '' iivbx ''

[Hormann 06]

[Porumbescu 05]

[Ju 05]

Tao Ju A general construction of coordinates Slide 4

Coordinates In A PolytopeCoordinates In A Polytope

• Points form vertices of a closed polytope

– x lies inside the polytope

• Example: A 2D triangle

– Unique (barycentric):

– Can be extended to any N-D simplex

• A general polytope

– Non-unique

– The triangle-trick can not be applied.

v

321

32

1vvv

vvx

A

Ab

321

32

1vvv

vvx

A

Ab

1v

x

2v3v

x

Tao Ju A general construction of coordinates Slide 5

Previous Work Previous Work

• 2D Polygons

– Wachspress [Wachspress 75][Loop 89][Meyer 02][Malsch 04]

• Barycentric within convex shapes

– Discrete harmonic [Desbrun 02][Floater 06]

• Homogeneous within convex shapes

– Mean value [Floater 03][Hormann 06]

• Homogeneous within any closed shape, barycentric within convex shapes and kernels of star-shapes

• 3D Polyhedrons and Beyond

– Wachspress [Warren 96][Ju 05]

– Discrete harmonic [Meyer 02]

– Mean value [Floater 05][Ju 05]

Tao Ju A general construction of coordinates Slide 6

Previous WorkPrevious Work

• A general construction in 2D [Floater 06]

– Complete: a single scheme that can construct all possible homogeneous coordinates in a convex polygon

– Reveals a simple connection between known coordinates via a parameter

• Wachspress:

• Mean value:

• Discrete harmonic:

• What about 3D and beyond? (this talk)

– To appear in Computer Aided Geometric Design (2007)

p

0p1p

2p

Tao Ju A general construction of coordinates Slide 7

To answer the question…To answer the question…

• Yes, a general construction exists in N-D

– A single scheme constructing all possible homogeneous coordinates in a convex simplicial polytope

• 2D polygons, 3D triangular polyhedrons, etc.

• The construction is geometric: coordinates correspond to some auxiliary shape

– Intrinsic geometric relation between known coordinates in N-D

• Wachspress: Polar dual

• Mean value: Unit sphere

• Discrete harmonic: Original polytope

– Easy to find new coordinates

Tao Ju A general construction of coordinates Slide 8

Homogenous WeightsHomogenous Weights

• We focus on an equivalent problem of finding weights such that

– Yields homogeneous coordinates by normalization

0)( xvw ii 0)( xvw ii

w

i

ii w

wb

i

ii w

wb

xiv

Tao Ju A general construction of coordinates Slide 9

2D Mean Value Coordinates2D Mean Value Coordinates

• We start with a geometric construction of 2D MVC

1. Place a unit circle at . An edge projects to an arc on the circle.

2. Write the integral of outward unit normal of each arc, , using the two vectors:

3. The integral of outward unit normal over the whole circle is zero. So the following weights are homogeneous:

x },{ 21 vvT T

Tr

)()( 2211 xvuxvur TTT )()( 2211 xvuxvur TTT

TvT

Tii

i

uw:

TvT

Tii

i

uw:

v1v2

x

Tr T

T

Tao Ju A general construction of coordinates Slide 10

2D Mean Value Coordinates2D Mean Value Coordinates

• To obtain :

– Apply Stoke’s Theorem

Tr

v1v2

x

Tr T

T

-n2T -n1

T

TTT nnr 21 TTT nnr 21

Tao Ju A general construction of coordinates Slide 11

Our General ConstructionOur General Construction

• Instead of a circle, pick any closed curve

1. Project each edge of the polygon onto a curve segment on .

2. Write the integral of outward unit normal of each arc, , using the two vectors:

3. The integral of outward unit normal over any closed curve is zero (Stoke’s Theorem). So the following weights are homogeneous:

},{ 21 vvT T

Tr

)()( 2211 xvuxvur TTT )()( 2211 xvuxvur TTT

TvT

Tii

i

uw:

TvT

Tii

i

uw:

v1v2

x

TrT

T

G

G

Tao Ju A general construction of coordinates Slide 12

Our General ConstructionOur General Construction

• To obtain :

– Apply Stoke’s Theorem

Tr

TTT ndndr 2211 TTT ndndr 2211

v1

x

d1 d2

v2 v1

xd1

-d2

v2T T

T

T

Tr Tr

Tao Ju A general construction of coordinates Slide 13

ExamplesExamples

• Some interesting result in known coordinates

– We call the generating curve

G

G

Wachspress(G is the polar dual)

Mean value(G is the unit circle)

Discrete harmonic(G is the original polygon)

Tao Ju A general construction of coordinates Slide 14

General Construction in 3DGeneral Construction in 3D

• Pick any closed generating surface

1. Project each triangle of the polyhedron onto a surface patch on .

2. Write the integral of outward unit normal of each patch, , using three vectors:

3. The integral of outward unit normal over any closed surface is zero. So the following weights are homogeneous:

},,{ 321 vvvT T

Tr

3

1)(

i iTi

T xvur

3

1)(

i iTi

T xvur

TvT

Tii

i

uw:

TvT

Tii

i

uw:

G

G

v2

v3

v1

x

T TrT

Tao Ju A general construction of coordinates Slide 15

-n3T

d1,2

General Construction in 3DGeneral Construction in 3D

• To obtain :

– Apply Stoke’s Theorem

Tr

3

11,1

i

Tiii

T ndr

3

11,1

i

Tiii

T ndr v2

v3

v1

x

T TrT

Tao Ju A general construction of coordinates Slide 16

ExamplesExamples

Wachspress(G: polar dual)

Mean value(G: unit sphere)

Discrete harmonic(G: the polyhedron)

Voronoi(G: Voronoi cell)

Tao Ju A general construction of coordinates Slide 17

An Equivalent Form – 2DAn Equivalent Form – 2D

• Same as in [Floater 06]

– Implies that our construction reproduces all homogeneous coordinates in a convex polygon

vi

vi-1 vi+1

x

Bi

vi

vi+1

x

Ai-1 Aivi-1ii

ii

i

i

i

ii AA

Bc

A

c

A

cw

1

1

1

1

ii

ii

i

i

i

ii AA

Bc

A

c

A

cw

1

1

1

1

where

2/|| xvdc iii 2/|| xvdc iii

Tao Ju A general construction of coordinates Slide 18

An Equivalent Form – 3DAn Equivalent Form – 3D

• 3D extension of [Floater 06]

– We showed that our construction yields all homogeneous coordinates in a convex triangular polyhedron.

j jj

jji

j j

jji AA

Bc

A

cw

1

,1,

j jj

jji

j j

jji AA

Bc

A

cw

1

,1,

where

6/|)()(|,, xvxvdc jijiji 6/|)()(|,, xvxvdc jijiji

vi

vj-1 vj+1

vj

x

Aj-1 Aj

vi

vj-1 vj+1

vj

x

Bj

Cj

Tao Ju A general construction of coordinates Slide 19

SummarySummary

• For any convex simplicial polytope

• Geometric

– Every closed generating shape (curve/surface/hyper-surface) yields a set of homogeneous coordinates

• Wachspress: polar dual

• Mean value: unit sphere

• Discrete harmonic: original polytope

• Voronoi: voronoi cell

• Complete

– Every set of homogeneous coordinates can be constructed by some generating shape

Tao Ju A general construction of coordinates Slide 20

Open QuestionsOpen Questions

• What about non-convex shapes?

– Coordinates may not exist along extension of faces

• Do we know other coordinates that are well-defined for non-convex, besides MVC?

• What about continuous shapes?

– General constructions known [Schaefer 07][Belyaev 06]

– What is the link between continuous and discrete constructions?

• What about non-simplicial polytopes?

– Initial attempt by [Langer 06]. Does such construction agree with the continuous construction?

• What about on a sphere?

– Initial attempt by [Langer 06], yet limited to within a hemisphere.