Generalized Barycentric Coordinates

Generalized Barycentric Coordinates


Anisimov Dmitry

Simple


Anisimov Dmitry

Simplex


Anisimov Dmitry

Simplex


Anisimov Dmitry

V1

V3 V2

P A2

A1

A3

A=A1+A2+A3

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


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


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.


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


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


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


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



Ai-1

Ai vi

vi+1

vi-1

P

Bi



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 .



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∑






•  Reproduc:on:

YES for

biv

i= P

i =1

n∑

bi=1

i =1

n∑

I.e.






•  Reproduc:on:

YES for

bi=1

i =1

n∑

biv

i= P

i =1

n∑

I.e.






•  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.



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






•  Reproduc:on:

bi=1

i =1

n∑

biv

i= P

i =1

n∑

YES for I.e.



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



Three-‐point coordinates:

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

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



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



Proper:es of Wachspress coordinates:

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


•  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



Proper:es of Wachspress 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



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


•  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

#$%

&%



Proper:es of Mean Value coordinates:


Education

Generalized Barycentric Coordinates