Generalized Barycentric Coordinates

Generalized  Barycentric  Coordinates  

Generalized  Barycentric  Coordinates  

Anisimov  Dmitry  

Simple  

Generalized  Barycentric  Coordinates  

Anisimov  Dmitry  

Simplex  

Generalized  Barycentric  Coordinates  

Anisimov  Dmitry  

Simplex  

Generalized  Barycentric  Coordinates  

Anisimov  Dmitry  

V1

V3 V2

P A2

A1

A3

A=A1+A2+A3

b1=A1/A b2=A2/A b3=A3/A

Generalized  Barycentric  Coordinates  

Anisimov  Dmitry  

1790-1868

1827

V1

V3 V2

P A2

A1

A3

A=A1+A2+A3

b1=A1/A b2=A2/A b3=A3/A

Generalized  Barycentric  Coordinates  

Anisimov  Dmitry  

V1

V3 V2

P

b1=A1/A b2=A2/A b3=A3/A

Proper:es:  

•  P is  inside  the  triangle  if  and  only  if  0 < b1, b2, b3 < 1.

Ø  If  b1, b2, b3 > 0 hence  P -­‐ within  the  interior  of  the  triangle.  

Ø  If  one  of  bi = 0 hence  P -­‐  on  some  edge  of  the  triangle.  

Ø  If  two  of  bi = 0 hence  P -­‐  in  some  vertex  of  the  triangle.  

Ø  b1 + b2 + b3 = 1.

Generalized  Barycentric  Coordinates  

Anisimov  Dmitry  

Proper:es:  

•  By  changing  the  values  of  b1, b2, b3 between  0  and  1,  the  point  P      will  move  smoothly  inside  the  triangle.  

V1

V3 V2

P

b1=A1/A b2=A2/A b3=A3/A

Generalized  Barycentric  Coordinates  

Anisimov  Dmitry  

Proper:es:  

•  P is  the  barycenter  of  the  points  v1, v2 and  v3 with  weights  A1, A2 and  A3 if  and  only  if:  

  P =

•  The  center  of  the  triangle  is  obtained  when  b1 = b2 = b3 = .

A1v1+A2v2+A3v3

A1+A2+A3

13

V1

V3 V2

P

b1=A1/A b2=A2/A b3=A3/A

Generalized  Barycentric  Coordinates  

Anisimov  Dmitry  

Proper:es:  

•  P is  inside  the  triangle  if  and  only  if  0 < b1, b2, b3 < 1.

•  By  changing  the  values  of  b1, b2, b3 between  0  and  1,  the  point  P  will  move  smoothly  inside  the  triangle.  

•  P is  the  barycenter  of  the  points  v1, v2 and  v3 with  weights  A1, A2 and  A3 if  and  only  if:  

P =  

A1v1+A2v2+A3v3

A1+A2+A3

V1

V3 V2

P

b1=A1/A b2=A2/A b3=A3/A

Generalized  Barycentric  Coordinates  

Anisimov  Dmitry  

Applica:ons:  •  Since  P is  inside  the  triangle  if  and  only  if  0 < b1, b2, b3 < 1

we  can  determine  if  a  point  P  is  inside  the  triangle.    •  Since  all  bi are  linear  polynomials  and  by  changing  the  values  of  b1, b2, b3 between  0  

and  1,  the  point  P  moves  smoothly  inside  the  triangle  

we  can  linearly  interpolate  data  placed  in  the  ver:ces  overall  triangle:    

F = bifi

i =1

3

∑

V1

V3 V2

P

b1=A1/A b2=A2/A b3=A3/A

Outline:  

•  Introduc:on  

•  Barycentric  Coordinates  for  Planar  Convex  Polygons  

Anisimov  Dmitry  

Barycentric  Coordinates  for  Planar  Convex  Polygons  

Anisimov  Dmitry  Generalized  Barycentric  Coordinates  

vi

vi+1

vi-1

P

Ai-1

Ai

Barycentric  Coordinates  for  Planar  Convex  Polygons  

Anisimov  Dmitry  Generalized  Barycentric  Coordinates  

Ai-1

Ai vi

vi+1

vi-1

P

Bi

Barycentric  Coordinates  for  Planar  Convex  Polygons  

Anisimov  Dmitry  Generalized  Barycentric  Coordinates  

Ai-1 Ai Bi

bi=

wi

wjj =1

n∑

Normalized  Barycentric  Coordinates:  

Where  weights:   wi=c

i +1A

i −1−c

iB

i+c

i −1A

i

Ai −1

Ai

with  certain  real  func:ons  ci .  

Barycentric  Coordinates  for  Planar  Convex  Polygons  

Anisimov  Dmitry  Generalized  Barycentric  Coordinates  

Do  bi sa:sfy  all  three  proper:es  of  triangular  barycentric  coordinates?  

I.e.  

•  Posi:vity: bi ≥ 0 for  all  i    

•  Par::on  of  unity:    

•  Reproduc:on:    

bi=1

i =1

n∑

biv

i= P

i =1

n∑

Barycentric  Coordinates  for  Planar  Convex  Polygons  

Anisimov  Dmitry  Generalized  Barycentric  Coordinates  

Do  bi sa:sfy  all  three  proper:es  of  triangular  barycentric  coordinates?  

•  Posi:vity: bi ≥ 0 for  all  i    

•  Par::on  of  unity:    

•  Reproduc:on:    

YES for

biv

i= P

i =1

n∑

bi=1

i =1

n∑

I.e.  

Barycentric  Coordinates  for  Planar  Convex  Polygons  

Anisimov  Dmitry  Generalized  Barycentric  Coordinates  

Do  bi sa:sfy  all  three  proper:es  of  triangular  barycentric  coordinates?  

•  Posi:vity: bi ≥ 0 for  all  i    

•  Par::on  of  unity:    

•  Reproduc:on:    

YES for

bi=1

i =1

n∑

biv

i= P

i =1

n∑

I.e.  

Barycentric  Coordinates  for  Planar  Convex  Polygons  

Anisimov  Dmitry  Generalized  Barycentric  Coordinates  

Do  bi sa:sfy  all  three  proper:es  of  triangular  barycentric  coordinates?  

•  Posi:vity: bi ≥ 0 for  all  i    

•  Par::on  of  unity:    

•  Reproduc:on:    

bi=1

i =1

n∑

biv

i= P

i =1

n∑

YES for

To  get  Posi:vity  we  have  to  properly  choose  func:ons  ci .  

I.e.  

Barycentric  Coordinates  for  Planar  Convex  Polygons  

Anisimov  Dmitry  Generalized  Barycentric  Coordinates  

We  choose  func:ons  ci to  be  Euclidean  distance  between  P and  vi to  the  power  k    :  

ci = rik with  ri = ||P - vi|| and    

vi

P

ri

k ∈ R

Barycentric  Coordinates  for  Planar  Convex  Polygons  

Anisimov  Dmitry  Generalized  Barycentric  Coordinates  

Do  bi sa:sfy  all  three  proper:es  of  triangular  barycentric  coordinates?  

•  Posi:vity: bi ≥ 0 for  all  i    

•  Par::on  of  unity:    

•  Reproduc:on:    

bi=1

i =1

n∑

biv

i= P

i =1

n∑

YES for I.e.  

Barycentric  Coordinates  for  Planar  Convex  Polygons  

Anisimov  Dmitry  Generalized  Barycentric  Coordinates  

With  such  a  choice  of  ci we  get  a  whole  family  of      three-­‐point  coordinates  bi :  

bi=

wi

wjj =1

n∑

with   wi=ri +1k A

i −1−r

ikB

i+r

i −1k A

i

Ai −1

Ai

Bi Ai-1

Ai ri-1 ri

ri+1

Barycentric  Coordinates  for  Planar  Convex  Polygons  

Anisimov  Dmitry  Generalized  Barycentric  Coordinates  

Three-­‐point  coordinates:  

•  Wachspress  Coordinates  for  k = 0    and  ci = 1.  

•  Mean  Value  Coordinates  for  k = 1 and  ci = ri  .  

Barycentric  Coordinates  for  Planar  Convex  Polygons  

Anisimov  Dmitry  Generalized  Barycentric  Coordinates  

Wachspress  coordinates:  For  the  first  :me  they  were  introduced  by  E.  L.  Wachspress  in  the  work:    “A  Ra:onal  Finite  Element  Basis”  in  1975.  

Weight  func:ons:   wi=

Di

Ai −1

Ai

and   bi=

wi

wjj =1

n∑

Di Ai-1 Ai

Barycentric  Coordinates  for  Planar  Convex  Polygons  

Anisimov  Dmitry  Generalized  Barycentric  Coordinates  

Proper:es  of  Wachspress  coordinates:  

•  Affine  precision:    •  Lagrange  property:    •  Smoothness:  bi are  C∞  inside  arbitrary  polygons*  

•  Par::on  of  unity:    

•  Behavior:  bi are  well-­‐defined  inside  convex  polygons  

•  Posi:vity:  bi are  posi:ve  inside  convex  polygons  

biϕ(v

i)

i =1

n∑ = ϕ for  any  affine  func:on    ϕ : R2 →Rd

bi(v

j) = δ

i , j=

1,i = j0,i ≠ j

#$%

&%

bi=1

i =1

n∑

*Except  poles  

Barycentric  Coordinates  for  Planar  Convex  Polygons  

Anisimov  Dmitry  Generalized  Barycentric  Coordinates  

Proper:es  of  Wachspress  coordinates:  

Barycentric  Coordinates  for  Planar  Convex  Polygons  

Anisimov  Dmitry  Generalized  Barycentric  Coordinates  

Mean  Value  coordinates:  For  the  first  :me  they  were  introduced  by  M.  Floater  in  the  work:    “Mean  Value  Coordinates”  in  2003.  

Weight  func:ons:  wi=ri −1

Ai−r

iB

i+r

i +1A

i −1

Ai −1

Ai

and   bi=

wi

wjj =1

n∑

Bi Ai-1

Ai ri-1 ri

ri+1

Barycentric  Coordinates  for  Planar  Convex  Polygons  

Anisimov  Dmitry  Generalized  Barycentric  Coordinates  

Proper:es  of  Mean  Value  coordinates:  

•  Affine  precision:  

•  Lagrange  property:  

•  Smoothness:  bi are  C∞  inside  arbitrary  polygons  except  at  the  ver:ces  vj where  they  are  only  C0  

•  Par::on  of  unity:    

•  Behavior:  bi are  well-­‐defined  inside  arbitrary  polygons    •  Posi:vity:  bi are  posi:ve  inside  convex  polygons  

biϕ(v

i)

i =1

n∑ = ϕ for  any  affine  func:on    ϕ : R2 →Rd

bi=1

i =1

n∑

bi(v

j) = δ

i , j=

1,i = j0,i ≠ j

#$%

&%

Barycentric  Coordinates  for  Planar  Convex  Polygons  

Anisimov  Dmitry  Generalized  Barycentric  Coordinates  

Proper:es  of  Mean  Value  coordinates:  

Generalized  Barycentric  Coordinates