1 Dr. Scott Schaefer Generalized Barycentric Coordinates.

1

Dr. Scott Schaefer

Generalized Barycentric Coordinates

2/95

Barycentric Coordinates

Given find weights such that

are barycentric coordinates

iw

i i

i ii

w

pwv

v

j j

i

w

w

v

1p

3p2p

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iw

i i

i ii

w

pwv

v

j j

i

w

w

v

1p

3p2p

v

w

w

w

ppp

3

2

1

321

Homogenous coordinates

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iw

i i

i ii

w

pwv

v

j j

i

w

w

v

1p

3p2p

vppp

w

w

w1

321

3

2

1

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iw

i i

i ii

w

pwv

v

j j

i

w

w

v

1p

3p2p

1A

2A3A

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iw

i i

i ii

w

pwv

v

j j

i

w

w

v

1p

3p2p

1A

2A3Aii Aw

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are barycentric coordinates 1p

3p

2p

4p

5p

v

iw

i i

i ii

w

pwv

v

j j

i

w

w

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Boundary Value Interpolation

Given , compute such that

Given values at , construct a function

Interpolates values at vertices Linear on boundary Smooth on interior

1f

2f 3f

4f

5f

i i

i ii

w

fwvf )(ˆ

i i

i ii

w

pwv

iw

if ip

ip

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Boundary Value Interpolation

Given , compute such that

Given values at , construct a function

Interpolates values at vertices Linear on boundary Smooth on interior

i i

i ii

w

fwvf )(ˆ

1f

2f 3f

4f

5f

i i

i ii

w

pwv

ipiw

if ip

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Multi-Sided Patches

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Multi-Sided Patches

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Multi-Sided Patches

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Multi-Sided Patches

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Multi-Sided Patches

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Multi-Sided Patches

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Multi-Sided Patches

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Multi-Sided Patches

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Multi-Sided Patches

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Wachspress Coordinates

v

ip

i

ii pwv

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v

i

ii pwv

ip

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),,( 1 iii ppvarea

v

i

ii pwv

ip

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),,( 1 iii ppvarea

niiiiii pppareaw ......),,( 122111

v

i

ii pwv

ip

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v

ii

iii

w

pwv

),,( 1 iii ppvarea

niiiiii pppareaw ......),,( 122111

ip

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v

ii

iii

w

pwv

),,( 1 iii ppvarea

ii

iiii

pppareaw

1

11 ),,(

ip

Polar Duals of Convex Polygons

Given a convex polyhedron P containing the origin, the polar dual is

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PxyxyPd 1|][

Properties of Polar Duals

is dual to a face with plane equation Each face with normal and vertex is

dual to the vertex

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ip 1xpi

in ip

ii

i

pn

n

in ip

ip

1

ii

i

pn

n

Properties of Polar Duals

is dual to a face with plane equation Each face with normal and vertex is

dual to the vertex

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ip 1xpi

ip

ip

ip

1

in

ii

i

pn

n

in

ii

i

pn

n

Coordinates From Polar Duals

Given a point v, translate v to origin Construct polar dual

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in ip

v



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vpi

vpi 1

in

)( vpn

n

ii

i



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vpi

vpi 1

0)()( 1

1

i i

i

ii

i

ii

i

vp

vp

vpn

n

vpn

n in

)( vpn

n

ii

i



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vpi

vpi 1

0)()( 1

1

i i

i

ii

i

ii

i

vp

vp

vpn

n

vpn

n in

)( vpn

n

ii

i



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vpi

vpi 1

vpvpn

nvpn

nw i

ii

i

ii

ii

)()( 1

1 in

)( vpn

n

ii

i



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vpi )( 1ii pp

)()(

)(

1

1

vppp

pp

iii

ii

vpvpn

nvpn

nw i

ii

i

ii

ii

)()( 1

1



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vpi )( 1ii pp

)()(

)(

1

1

vppp

pp

iii

ii

)()(

)(

)()(

)(

1

1

1

1

vppp

pp

vppp

ppw

iii

ii

iii

iii



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ip )( 1ii pp

v

1ip 1ip)()(

)(

)()(

)(

1

1

1

1

vppp

pp

vppp

ppw

iii

ii

iii

iii



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ip )( 1ii pp

),,(

)(

),,(

)(

1

1

1

1

ii

ii

ii

iii ppvarea

pp

ppvarea

ppw

v

1ip 1ip



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ip )( 1ii pp

),,(

)(

),,(

)(

1

1

1

1

ii

ii

ii

iii ppvarea

pp

ppvarea

ppw

v

1ip 1ip



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ip )( 1ii pp

),,(),,(

),,(

11

11

iiii

iiii ppvareappvarea

pppareaw

v

1ip 1ip

Identical to Wachspress Coordinates!

Extensions into Higher Dimensions

Compute polar dual Volume of pyramid from dual face to origin

is barycentric coordinate

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Wachspress Coordinates – Summary

Coordinate functions are rational and of low degree

Coordinates are only well-defined for convex polygons

wi are positive inside of convex polygons

3D and higher dimensional extensions (for convex shapes) do exist

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Mean Value Coordinates

i i

i ii

w

pwv

1p

2p

3p 4p

5p

v

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1p

2p

3p 4p

5p

v

0)( i ii vpw

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1p

2p

3p 4p

5p

v vS

0)( i ii vpw

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1p

2p

3p 4p

5p

im

v vS

0)( i ii vpw

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1p

2p

3p 4p

5p

im

v vS

0i

im

0)( i ii vpw

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1p

2p

3p 4p

5p

im

v vS

0i

im

)()( 11 vpvpm iiiii

0)( i ii vpw

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1p

2p

3p 4p

5p

im

v vS

0))(( i

iii vp

0i

im

)()( 11 vpvpm iiiii

0)( i ii vpw

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im

v1n

2n

0)( i ii vpw

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Apply Stokes’ Theorem

im

v1n

2n021 nnmi

0)( i ii vpw

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Comparison

convex polygons(Wachspress Coordinates)

closed polygons(Mean Value Coordinates)

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Comparison



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Comparison



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Comparison



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3D Mean Value Coordinates

v

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Exactly same as 2D but must compute mean vector for a given spherical triangle


v

m

m

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Build wedge with face normals

v

m

kn

kn

m

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Build wedge with face normals Apply Stokes’ Theorem,


v

m

kn

k

kn

03

12

1

mnk

kk

m

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Deformations using Barycentric Coordinates

i i

i ii

w

pwv

ipv

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i i

i ii

w

pwv

ipv

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i i

i ii

w

pwv

ˆˆ

ip̂

i i

i ii

w

pwv

ipv

v̂

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i i

i ii

w

pwv

vip

i i

i ii

w

pwv

ˆˆ

v̂

ip̂

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Deformation Examples

Control Mesh Surface Computing Weights Deformation

216 triangles 30,000 triangles 1.9 seconds 0.03 seconds

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Real-time!

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Mean Value Coordinates – Summary

Coordinate functions are NOT rational Coordinates are only well-defined for any

closed, non-self-intersecting polygon/surface wi are positive inside of convex polygons, but

not in general

Constructing a Laplacian Operator

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0)()(2 vFvF


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0)()(2 vFvF

Laplacian


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2

21 )(max vF

F

Euler-Lagrange Theorem


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0p

1p

2p

3p

4p

5p

2

21 )(max vF

F


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0f

1f

2f

3f

4f

5f

2

21 )(max vF

F


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0f

1f

2f

3f

4f

5f

v

2

21 )(max vF

F


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v

),( 00 fp

),( 11 fp

),( 22 fp

2

21 )(max vF

F


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),( 00 fp

),( 11 fp

),( 22 fp

3

121),,(

1 ),,()(210

iiiippparea fvppareavF

v

2

21 )(max vF

F


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v

),( 00 fp

),( 11 fp

),( 22 fp

3

121),,(

1 ),,()(210


2

21 )(max vF

F


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v

),( 00 fp

),( 11 fp

),( 22 fp

3

121),,(

1 ),,()(210


?),,( 21 vpparea

2

21 )(max vF

F


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v

),( 00 fp

),( 11 fp

),( 22 fp

3

121),,(

1 ),,()(210


)(),,( 2121 ppvpparea

2

21 )(max vF

F


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v

),( 00 fp

),( 11 fp

),( 22 fp

3

121),,(

1 )()(210

iiiippparea fppvF

2

21 )(max vF

F


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v

),( 00 fp

),( 11 fp

),( 22 fp

2

210102021),,(12

)()()()( 2210

fppfppfppvFppparea

2

21 )(max vF

F


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v

),( 00 fp

),( 11 fp

),( 22 fp

120210201021100221

22

2

102

1

2

022

0

2

21

),,(12

)()(2)()(2)()(2)( 2

210 ffppppffppppffpppp

fppfppfppvF

ppparea

2

21 )(max vF

F


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v

),( 00 fp

),( 11 fp

),( 22 fp

120210201021100221

22

2

102

1

2

022

0

2

21),,(

12

)()(2)()(2)()(2)(

210

ffppppffppppffpppp

fppfppfppvF ppparea

2

21 )(max vF

F


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v

),( 00 fp

),( 11 fp

),( 22 fp

21021102210

2

21),,(12

0

)()(2)()(22)(210

fppppfppppfppvFf ppparea

2

21 )(max vF

F


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v

),( 00 fp

),( 11 fp

),( 22 fp

21012120210

2

21),,(12

21

0

)()()()()(210

fppppfppppfppvFf ppparea

2

21 )(max vF

F


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v

),( 00 fp

),( 11 fp

),( 22 fp

211012

110121

22021

220210

210

2

212

21

0 )sin(

)cos(

)sin(

)cos(

),,()( f

pppp

ppppf

pppp

ppppf

ppparea

ppvF

f

0

1

2

2

21 )(max vF

F


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v

),( 00 fp

),( 11 fp

),( 22 fp

2112012

2

21

0

)cot()cot()cot()cot()( fffvFf

0

1

2

2

21 )(max vF

F


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2,1i1,i

2,i

),( ii fp

1,1i

01,12,1,12,0 )cot()cot()cot()cot()( ffpFi

iii

iii

),( 00 fp

Harmonic Coordinates

Solution to Laplace’s equation with boundary constraints

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0)( vF



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0)( vF



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0)( vF

1

00

00

0

0

0

0000

8.06.0

4.02.0

75.0

45.0



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0)( vF

1

00

00

0

0

0

0000

8.06.0

4.02.0

75.0

45.0

V

F

C

CL T 0

0



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0)( vF

1

00

00

0

0

0

0000

8.06.0

4.02.0

75.0

45.0

V

F

C

CL T 0

0

ith row contains laplacian for ith vertex



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0)( vF

v


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Harmonic Coordinates – Summary

Positive, smooth coordinates for all polygons Fall off with respect to geodesic distance, not

Euclidean distance Only approximate solutions exist and require

matrix solve whose size is proportional to accuracy

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Harmonic Coordinates – Summary

Positive, smooth coordinates for all polygons Fall off with respect to geodesic distance, not

Euclidean distance Only approximate solutions exist and require

matrix solve whose size is proportional to accuracy

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Barycentric Coordinates – Summary

Infinite number of barycentric coordinates Constructions exists for smooth shapes too Challenge is finding coordinates that are:

well-defined for arbitrary shapespositive on the interior of the shapeeasy to computesmooth

1 Dr. Scott Schaefer Generalized Barycentric Coordinates.

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Transcript of 1 Dr. Scott Schaefer Generalized Barycentric Coordinates.