a general, geometric construction of coordinates in any dimensions

Tao Ju A general construction of coordinates Slide 1

A general, geometric construction of coordinates in any dimensions

A general, geometric construction of coordinates in any dimensions

Tao Ju

Washington University in St. Louis

http://www.seas.wustl.edu/


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



ApplicationsApplications

• Value interpolation

– Color/Texture interpolation

• Mapping

– Shell texture

– “Cage-based” deformation

ii fbxf )( ii fbxf )(

iivbx '' iivbx ''

[Hormann 06]

[Porumbescu 05]

[Ju 05]



How to find coordinates?How to find coordinates?

iv

x

1, iii bwherevbx 1, iii bwherevbx



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



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]

• Barycentric within convex shapes, homogeneous within non-convex shapes

• 3D Polyhedrons and Beyond

– Wachspress [Warren 96][Ju 05a]

– Discrete harmonic [Meyer 02]

– Mean value [Floater 05][Ju 05b]



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

– Homogeneous coordinates parameterized by variable p

• Wachspress: p=-1

• Mean value: p=0

• Discrete harmonic: p=1

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

– Tao Ju, Peter Liepa, Joe Warren, CAGD 2007



The Short AnswerThe Short Answer

• Yes, such general construction exists in N-D

– Complete: A single scheme constructing all possible homogeneous coordinates

• Applicable to any convex simplicial polytope

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

– Geometric: The coordinates are parameterized by an auxiliary shape

• Wachspress: Polar dual

• Mean value: Unit sphere

• Discrete harmonic: Original polytope



The Longer Answer…The Longer Answer…

• 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



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




• To obtain :

– Apply Stoke’s Theorem

Tr

v1v2

x

Tr T

T

-n2T -n1

T

TTT nnr 21 TTT nnr 21




• We start with a geometric construction of 2D MVC

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


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

x },{ 21 vvT T

Tr


TvT

Tii

i

uw:

TvT

Tii

i

uw:

v1v2

x

Tr T

T



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 .


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


TvT

Tii

i

uw:

TvT

Tii

i

uw:

v1v2

x

TrT

T

G

G



Our General ConstructionOur General Construction

• To obtain :


Tr

TTT ndndr 2211 TTT ndndr 2211

v1

x

d1 d2

v2 v1

xd1

-d2

v2T T

T

T

Tr Tr



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)



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



-n3T

d1,2

General Construction in 3DGeneral Construction in 3D

• To obtain :


Tr

3

11,1

i

Tiii

T ndr

3

11,1

i

Tiii

T ndr v2

v3

v1

x

T TrT



ExamplesExamples

Wachspress(G: polar dual)

Mean value(G: unit sphere)

Discrete harmonic(G: the polyhedron)

Voronoi(G: Voronoi cell)



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



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



SummarySummary

• Homogenous coordinates construction in any dimensions

– Complete: Reproduces all homogeneous coordinates in a convex simplicial polytope

– Geometric: Each set of homogeneous coordinates is constructed from a closed generating shape (curve/surface/hyper-surface)

• Polar dual: Wachspress coordinates

• Unit sphere: Mean value coordinates

• Original polytope: Discrete harmonic coordinates

• Voronoi cell: Voronoi coordinates



Open QuestionsOpen Questions

• What about continuous shapes?

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

– What is the link between continuous and discrete constructions?

• Discrete coordinates equivalent to continuous coordinates when applied to a piece-wise linear shape

• What about non-convex shapes?

– Are there coordinates well-defined for non-convex shapes, besides MVC?

• What about non-simplicial polytopes?

– Are there closed-form MVC for non-triangular meshes that agree with the continuous definition of MVC?

• What about on a sphere?

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


a general, geometric construction of coordinates in any dimensions

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