Barycentric Coordinates

전체 글

(1)

Barycentric

Coordinates

Wanho Choi

(wanochoi.com)

(2)

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 condition

(3)

Vector + &

-•

Component-wise operation

A

B

A

+ B

−B

A

− B

Preliminaries

(4)

Dot Product

Projection

A

= (a

x

,a

y

,a

z

)

B

= (b

x

,b

y

,b

z

)

A

⋅B ≡ a

x

b

x

+ a

y

b

y

+ a

z

b

z

= abcos

θ

θ

a

b

A

proj Preliminaries

(5)

Simplex

A generalization of the notion

of a triangle or tetrahedron to arbitrary dimensions

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

(6)

The Area

Preliminaries

≡ 1 cm

2

= 100 mm

2 promise

=

4 cm

2 cm

= 1 cm

2

× 8 EA = 8 cm

2

(7)

The Area of a Triangle

(8)

2x2 Matrix Inverse

Preliminaries

(9)

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 . 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

[

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

(10)

Quiz

Where is ? And, explain the barycentric

coordinates.

p

0

p

1

p

2

p

=

1

6

p

0

+

1

3

p

1

+

1

2

p

2

p

(11)

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