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# Shape Matching

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(1)

## Shape Matching

(2)

i

i

### )

(translation, rotation, and isotropic scale)

(3)

(4)

(5)

(6)

(7)

i

i 2 i=1 n

(8)

i

i 2 i=1 n

1

1 2

2

2 2

n

n 2

## ∑

(9)

i

i 2 i=1 n

1

1 2

2

2 2

n

n 2

### ⎦

∵ ∂ ∂A Api − qi 2 ⎡ ⎣ ⎤⎦ = 2 Ap

i − qi

pi T ⎛ ⎝⎜ ⎞⎠⎟

1

1

1T

2

2

2T

n

n

nT

## ∑

(10)

i

i 2 i=1 n

1

1 2

2

2 2

n

n 2

### ⎦

∵ ∂ ∂A Api − qi 2 ⎡ ⎣ ⎤⎦ = 2 Ap

i − qi

pi T ⎛ ⎝⎜ ⎞⎠⎟

1

1

1T

2

2

2T

n

n

nT

1

1T

1

1T

2

2T

2

2T

n

nT

n

nT

## ∑

(11)

i

i 2 i=1 n

1

1 2

2

2 2

n

n 2

### ⎦

∵ ∂ ∂A Api − qi 2 ⎡ ⎣ ⎤⎦ = 2 Ap

i − qi

pi T ⎛ ⎝⎜ ⎞⎠⎟

1

1

1T

2

2

2T

n

n

nT

1

1T

1

1T

2

2T

2

2T

n

nT

n

nT

i

iT i=1 n

i

i T i=1 n

(12)

## ∑

i

i 2 i=1 n

1

1 2

2

2 2

n

n 2

### ⎦

∵ ∂ ∂A Api − qi 2 ⎡ ⎣ ⎤⎦ = 2 Ap

i − qi

pi T ⎛ ⎝⎜ ⎞⎠⎟

1

1

1T

2

2

2T

n

n

nT

1

1T

1

1T

2

2T

2

2T

n

nT

n

nT

i

iT i=1 n

i

i T i=1 n

(13)

i

i 2 i=1 n

1

1 2

2

2 2

n

n 2

### ⎦

∵ ∂ ∂A Api − qi 2 ⎡ ⎣ ⎤⎦ = 2 Ap

i − qi

pi T ⎛ ⎝⎜ ⎞⎠⎟

1

1

1T

2

2

2T

n

n

nT

1

1T

1

1T

2

2T

2

2T

n

nT

n

nT

i

iT i=1 n

i

i T i=1 n

i

iT i=1 n

i

i T i=1 n

−1

## ∑

(14)
(15)

2

T

(16)

2

### p

T Let A= a b c d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥, p =xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥, and q= uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥.

(17)

2

### p

T Let A= a b c d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥, p =xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥, and q= uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥.

### For simplicity, let's just look at the 2D case.

When θ∈!, ∂θ ∂A ≡ ∂θ ∂a ∂θ ∂b ∂θ ∂c ∂θ ∂d ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ .

(18)

2

### p

T Let A= a b c d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥, p =xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥, and q= uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥.

### For simplicity, let's just look at the 2D case.

When θ∈!, ∂θ ∂A ≡ ∂θ ∂a ∂θ ∂b ∂θ ∂c ∂θ ∂d ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ . ∂ ∂A Ap− q 2 ⎡⎣ ⎤⎦

(19)

2

### p

T Let A= a b c d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥, p =xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥, and q= uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥.

### For simplicity, let's just look at the 2D case.

When θ∈!, ∂θ ∂A ≡ ∂θ ∂a ∂θ ∂b ∂θ ∂c ∂θ ∂d ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ . ∂ ∂A Ap− q 2 ⎡⎣ ⎤⎦ = ∂A∂ ⎡a bc d ⎤ ⎦ ⎥⎡ xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

(20)

2

### p

T Let A= a b c d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥, p =xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥, and q= uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥.

### For simplicity, let's just look at the 2D case.

When θ∈!, ∂θ ∂A ≡ ∂θ ∂a ∂θ ∂b ∂θ ∂c ∂θ ∂d ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ . ∂ ∂A Ap− q 2 ⎡⎣ ⎤⎦ = ∂A∂ ⎡a bc d ⎤ ⎦ ⎥⎡ xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂∂A ax+ by cx+ dy ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

(21)

2

### p

T Let A= a b c d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥, p =xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥, and q= uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥.

### For simplicity, let's just look at the 2D case.

When θ∈!, ∂θ ∂A ≡ ∂θ ∂a ∂θ ∂b ∂θ ∂c ∂θ ∂d ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ . ∂ ∂A Ap− q 2 ⎡⎣ ⎤⎦ = ∂A∂ ⎡a bc d ⎤ ⎦ ⎥⎡ xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂∂A ax+ by cx+ dy ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂Aax+ by − u cx+ dy − v ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

(22)

2

### p

T Let A= a b c d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥, p =xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥, and q= uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥.

### For simplicity, let's just look at the 2D case.

When θ∈!, ∂θ ∂A ≡ ∂θ ∂a ∂θ ∂b ∂θ ∂c ∂θ ∂d ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ . ∂ ∂A Ap− q 2 ⎡⎣ ⎤⎦ = ∂A∂ ⎡a bc d ⎤ ⎦ ⎥⎡ xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂∂A ax+ by cx+ dy ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂Aax+ by − u cx+ dy − v ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂ ∂A

ax+ by − u

2 + cx + dy − v

2 ⎡⎣ ⎤⎦

(23)

2

### p

T Let A= a b c d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥, p =xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥, and q= uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥.

### For simplicity, let's just look at the 2D case.

When θ∈!, ∂θ ∂A ≡ ∂θ ∂a ∂θ ∂b ∂θ ∂c ∂θ ∂d ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ . ∂ ∂A Ap− q 2 ⎡⎣ ⎤⎦ = ∂A∂ ⎡a bc d ⎤ ⎦ ⎥⎡ xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂∂A ax+ by cx+ dy ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂Aax+ by − u cx+ dy − v ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂ ∂A

ax+ by − u

2 + cx + dy − v

### )

2 ⎡⎣ ⎤⎦ = ∂ ∂a(ax+ by − u) 2 ∂ ∂b(ax+ by − u) 2 ∂ ∂c(cx+ dy − v) 2 ∂ ∂d(cx+ dy − v) 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥

(24)

2

### p

T Let A= a b c d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥, p =xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥, and q= uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥.

### For simplicity, let's just look at the 2D case.

When θ∈!, ∂θ ∂A ≡ ∂θ ∂a ∂θ ∂b ∂θ ∂c ∂θ ∂d ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ . ∂ ∂A Ap− q 2 ⎡⎣ ⎤⎦ = ∂A∂ ⎡a bc d ⎤ ⎦ ⎥⎡ xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂∂A ax+ by cx+ dy ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂Aax+ by − u cx+ dy − v ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂ ∂A

ax+ by − u

2 + cx + dy − v

### )

2 ⎡⎣ ⎤⎦ = ∂ ∂a(ax+ by − u) 2 ∂ ∂b(ax+ by − u) 2 ∂ ∂c(cx+ dy − v) 2 ∂ ∂d(cx+ dy − v) 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = 2(ax+ by − u)x 2(ax + by − u)y

2(cx+ dy − v)x 2(cx + dy − v)y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

(25)

2

### p

T Let A= a b c d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥, p =xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥, and q= uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥.

### For simplicity, let's just look at the 2D case.

When θ∈!, ∂θ ∂A ≡ ∂θ ∂a ∂θ ∂b ∂θ ∂c ∂θ ∂d ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ . ∂ ∂A Ap− q 2 ⎡⎣ ⎤⎦ = ∂A∂ ⎡a bc d ⎤ ⎦ ⎥⎡ xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂∂A ax+ by cx+ dy ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂Aax+ by − u cx+ dy − v ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂ ∂A

ax+ by − u

2 + cx + dy − v

### )

2 ⎡⎣ ⎤⎦ = ∂ ∂a(ax+ by − u) 2 ∂ ∂b(ax+ by − u) 2 ∂ ∂c(cx+ dy − v) 2 ∂ ∂d(cx+ dy − v) 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = 2(ax+ by − u)x 2(ax + by − u)y

2(cx+ dy − v)x 2(cx + dy − v)y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = 2

(ax+ by − u)x (ax + by − u)y (cx+ dy − v)x (cx + dy − v)y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

(26)

2

### p

T Let A= a b c d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥, p =xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥, and q= uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥.

### For simplicity, let's just look at the 2D case.

When θ∈!, ∂θ ∂A ≡ ∂θ ∂a ∂θ ∂b ∂θ ∂c ∂θ ∂d ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ . ∂ ∂A Ap− q 2 ⎡⎣ ⎤⎦ = ∂A∂ ⎡a bc d ⎤ ⎦ ⎥⎡ xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂∂A ax+ by cx+ dy ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂Aax+ by − u cx+ dy − v ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂ ∂A

ax+ by − u

2 + cx + dy − v

### )

2 ⎡⎣ ⎤⎦ = ∂ ∂a(ax+ by − u) 2 ∂ ∂b(ax+ by − u) 2 ∂ ∂c(cx+ dy − v) 2 ∂ ∂d(cx+ dy − v) 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = 2(ax+ by − u)x 2(ax + by − u)y

2(cx+ dy − v)x 2(cx + dy − v)y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = 2

(ax+ by − u)x (ax + by − u)y (cx+ dy − v)x (cx + dy − v)y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = 2 ax+ by − u cx+ dy − v ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥⎡⎣ x y ⎤⎦

(27)

2

### p

T Let A= a b c d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥, p =xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥, and q= uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥.

### For simplicity, let's just look at the 2D case.

When θ∈!, ∂θ ∂A ≡ ∂θ ∂a ∂θ ∂b ∂θ ∂c ∂θ ∂d ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ . ∂ ∂A Ap− q 2 ⎡⎣ ⎤⎦ = ∂A∂ ⎡a bc d ⎤ ⎦ ⎥⎡ xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂∂A ax+ by cx+ dy ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂Aax+ by − u cx+ dy − v ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂ ∂A

ax+ by − u

2 + cx + dy − v

### )

2 ⎡⎣ ⎤⎦ = ∂ ∂a(ax+ by − u) 2 ∂ ∂b(ax+ by − u) 2 ∂ ∂c(cx+ dy − v) 2 ∂ ∂d(cx+ dy − v) 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = 2(ax+ by − u)x 2(ax + by − u)y

2(cx+ dy − v)x 2(cx + dy − v)y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = 2

(ax+ by − u)x (ax + by − u)y (cx+ dy − v)x (cx + dy − v)y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = 2 ax+ by − u cx+ dy − v ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥⎡⎣ x y ⎤⎦ = 2 a b c d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥⎡ xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ x y

(28)

2

### p

T Let A= a b c d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥, p =xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥, and q= uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥.

### For simplicity, let's just look at the 2D case.

When θ∈!, ∂θ ∂A ≡ ∂θ ∂a ∂θ ∂b ∂θ ∂c ∂θ ∂d ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ . ∂ ∂A Ap− q 2 ⎡⎣ ⎤⎦ = ∂A∂ ⎡a bc d ⎤ ⎦ ⎥⎡ xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂∂A ax+ by cx+ dy ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂Aax+ by − u cx+ dy − v ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂ ∂A

ax+ by − u

2 + cx + dy − v

### )

2 ⎡⎣ ⎤⎦ = ∂ ∂a(ax+ by − u) 2 ∂ ∂b(ax+ by − u) 2 ∂ ∂c(cx+ dy − v) 2 ∂ ∂d(cx+ dy − v) 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = 2(ax+ by − u)x 2(ax + by − u)y

2(cx+ dy − v)x 2(cx + dy − v)y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = 2

(ax+ by − u)x (ax + by − u)y (cx+ dy − v)x (cx + dy − v)y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = 2 ax+ by − u cx+ dy − v ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥⎡⎣ x y ⎤⎦ = 2 a b c d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥⎡ xy ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥− uv ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ x y = 2 Ap − q

pT

(29)

(30)

(31)

(32)

(33)

(34)

(35)

## ∑

(36)
(37)

### Soft body

(38)
(39)

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