Barycentric
Coordinates
Wanho Choi
(wanochoi.com)
Barycentric Coordinates
•
A coordinate system
in which the location of a point of a simplex
C
A
B
P
w
A= ΔPBC
ΔABC
P = w
A× A + w
B× B + w
C× C
w
B= ΔPCA
ΔABC
w
C= ΔPAB
ΔABC
= ΔPBC
ΔABC
= ΔPBC
ΔABC
= ΔPBC
ΔABC
0 ≤ w
A, w
B, w
C≤ 1
w
A+ w
B+ w
C= 1
inside conditionVector + &
-•
Component-wise operation
A
B
A
+ B
−B
A
− B
PreliminariesDot Product
•
Projection
A
= (a
x,a
y,a
z)
B
= (b
x,b
y,b
z)
A
⋅B ≡ a
xb
x+ a
yb
y+ a
zb
z= abcos
θ
θ
a
b
A
proj PreliminariesSimplex
•
A generalization of the notion
of a triangle or tetrahedron to arbitrary dimensions
0-Simplex 1-Simplex 2-Simplex 3-Simplex
The Area
Preliminaries≡ 1 cm
2= 100 mm
2 promise=
4 cm
2 cm
= 1 cm
2× 8 EA = 8 cm
2The Area of a Triangle
2x2 Matrix Inverse
Preliminaries
αAB + βAC = P − A
[AB . x AC . xAB . y AC . y]
[
α
β]
=
[Q . xQ . y]Its barycentric coordinates:
wA = 1 − α − β wB = α wC = β A B C P α 1 − α β 1 − β D wB = ΔAPC
ΔABC = ΔADCΔABC = α
αAB . x + βAC . x = Q . x αAB . y + βAC . y = Q . y ⟺ αAB + βAC + A = P ⟺ αAB + βAC = Q ⟺ ⟺ AB = B − A AC = C − A Q = P − A A B C P α 1 − α β 1 − β
[
α
β]
=
[AB . x AC . xAB . y AC . y] −1 [Q . xQ . y] ⟺[
α
β]
=
det [1 −AB . y AB . x ][AC . y −AC . x Q . xQ . y] det = AB . x × AC . y − AC . x × AB . y⟺
α = (AC . y × Q . x − AC . x × Q . y)/det β = (AB . x × Q . y − AB . y × Q . x)/det
⟺
β ≥ 0 & β ≤ 1 & α ≥ 0 & α + β ≤ 1
