Generalized Barycentric Coordinates
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Transcript of 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