# Barycentric Coordinates

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A

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B

C

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inside condition

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Preliminaries

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## Dot Product

x

y

z

x

y

z

x

x

y

y

z

z

### A

proj Preliminaries

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## Simplex

### of a triangle or tetrahedron to arbitrary dimensions

0-Simplex 1-Simplex 2-Simplex 3-Simplex

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Preliminaries

2

2 promise

2

2

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## 2x2 Matrix Inverse

Preliminaries

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αAB + βAC = P − A

[AB . x AC . xAB . y AC . y]

### =

[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

αAB . x + βAC . x = Q . x αAB . y + βAC . y = Q . yαAB + βAC + A = PαAB + βAC = Q ⟺ ⟺ AB = B − A AC = C − A Q = P − A A B C P α 1 − α β 1 − β

### =

[AB . x AC . xAB . y AC . y] −1 [Q . xQ . y]

### =

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

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0

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0

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