( , ), i [0, N 1] p i t i p ( t ), t [ , ] t 0 t N 1 p ( ) = t i - PowerPoint PPT Presentation

( , ), i ∈ [0, N − 1] p i t i p ( t ), t ∈ [ , ] t 0 t N −1 p ( ) = t i p i

⎧ ⎪ p ( t ) = (1 − α ( t )) + α ( t ) p i p i +1 ⎨ ∀ t ∈ [ , ] , t − t i t i t i +1 ⎩ ⎪ α ( t ) = − t i +1 t i

N −1 d ( ) c j ∑ ∑ t i p ( t ) = i p j i =0 j =0 N −1 ∑ p ( t ) = ( t ) α i p i i =0 ( ) α i d

N −1 ∑ ∀ t ∈ [ , ] , p ( t ) = ( t ) ∀ i ∈ [0, N − 1] p ( ) = t 0 t N −1 α i p i t i p i i =0 N − 1 N −1 N −1 t − t k ∑ ∏ p ( t ) = ( t ) ( t ) = α i p i α i − t i t k i =0 k =0, k ≠ i ( ) = 1 ( ) = 0 α i t i α i t k

(p 5 ,t 5 ) (p 2 ,t 2 ) (p 1 ,t 1 ) (p 4 ,t 4 ) (p 0 ,t 0 ) (p 3 ,t 3 ) C 2

d 0 c 3 s 3 c 2 s 2 { p ( s ) = + + s + c 1 c 0 p 0 p 1 d 1 p ′ p ′ p (0) = , p (1) = , (0) = , (1) = p 0 p 1 d 0 d 1 ⇒ ⎧ ⎧ ⎪ ⎪ = p 0 = c 0 c 0 p 0 ⎪ ⎪ ⎪ ⎪ + + + = p 1 = c 3 c 2 c 1 c 0 c 1 d 0 ⎨ ⎨ ⇒ ⎪ ⎪ ⎪ = d 0 ⎪ = −3 + 3 − 2 − c 1 c 2 p 0 p 1 d 0 d 1 ⎩ ⎩ ⎪ ⎪ 3 + 2 + = d 1 = 2 − 2 + + c 3 c 2 c 1 c 3 p 0 p 1 d 0 d 1 s 2 p 1 s 2 d 1 s 3 s 2 s 3 s 2 s 3 s 3 ∀ s ∈ [0, 1] , p ( s ) = (2 − 3 + 1) + ( − 2 + s ) + (−2 + 3 ) + ( − ) p 0 d 0 t − t i t ∈ [ , ] s = t i t i +1 − t i +1 t i

− p i +1 p i −1 = μ d i − t i +1 t i −1 ∈ [0, 2] μ μ = 1

p ( t ) ( p i t i ) i ∈[0, N −1] , t ∈ [ , ] t 1 t N −2 t ∈ [ , ] i t i t i +1 − − p i +1 p i −1 p i +2 p i = μ = μ d i d i +1 − − t i +1 t i −1 t i +2 t i s 2 p i +1 s 2 d i +1 s 3 s 2 s 3 s 2 s 3 s 3 p ( t ) = (2 − 3 + 1) + ( − 2 + s ) + (−2 + 3 ) + ( − ) p i d i t − t i s = − t i +1 t i

C 1 C 2 C 2

p i ∑ k ω k b i = k p = B ω p = + ( − ) ∑ k b 0 b k b 0      d k p = b 0 + D ω ⇒ D ⇒

( x , y , z ) ( , , ) s x s y s z ( , , , ) q x q y q z q w p ( t ) = f ( w ( t )) w w w w t t t

e i θ 1 → = R 1 z 1 e i θ 2 → = R 2 z 2 e i ( α +(1− α ) ) R α θ 1 θ 2 → z = 1→2

⎧ ⎛ ⎞ ⎪ r 00 r 01 r 02 ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ R = r 10 r 11 r 12 ⎝ ⎠ ⎨ v ′ r 20 r 21 r 22 = R v ⎪ ⎪ ⎪ ⎪ R = R 1 R 2 ⎪ ⎩ ⎪ R T R = Id M = (1 − α ) R 1 + α R 2 M

( x , y , z ) ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ cos( θ ) sin( θ ) 0 cos( θ ) 0 sin( θ ) 1 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = = = 0 cos( θ ) sin( θ ) R x − sin( θ ) cos( θ ) 0 R y 0 1 0 R z ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0 0 1 − sin( θ ) 0 cos( θ ) 0 − sin θ ) cos( θ )

n θ u 1 u 2 n = ( × )/∥ × ∥ u 1 u 2 u 1 u 2 θ = acos( ⋅ ) u 1 u 2 u 2

( n , θ ) v v = + v ∥ v ⊥ v ′ v ′ v ′ = + ⊥ ∥ v ′ ⇒ = + (cos( θ ) + sin( θ ) n × ) v ∥ v ⊥ v ⊥ = ( v ⋅ n ) n v ∥ = v − ( v ⋅ n ) n v ⊥ v ′ = ( v ⋅ n ) n + cos( θ )( v − ( v ⋅ n ) n ) + sin( θ ) n × ( v − ( v ⋅ n ) n ) v ′ ⇒ = cos( θ ) v + sin( θ ) n × v + (1 − cos( θ )) ( v ⋅ n ) n

v ′ = cos( θ ) v + sin( θ ) n × v + (1 − cos( θ )) ( v ⋅ n ) n = R ( n , θ ) v ⎛ ⎞ ⎛ ⎞ n x n x ⎜ ⎟ ⎜ ⎟ v ′ = cos( θ ) v + sin( θ ) × v + (1 − cos( θ )) ( ) v n y n x n y n z n y ⎝ ⎠ ⎝ ⎠ n z n z n 2 ⎛ ⎞ ⎛ ⎞ 0 − n z n x n y n x n z n y x ⎜ ⎟ ⎜ ⎟ v ′ n 2 = cos( θ ) v + sin( θ ) v + (1 − cos( θ )) 0 − n x n x n y n y n z v n z y ⎝ ⎠ ⎝ ⎠ n 2 − n y 0 n x n x n z n y n z          z  K K 2 n 2 ⎛ ⎞ cos( θ ) + (1 − cos( θ )) (1 − cos( θ )) − sin( θ ) (1 − cos( θ )) + sin( θ ) n x n y n z n x n z n y x ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ v ′ n 2 ⎜ ⎟ = (1 − cos( θ )) + sin( θ ) cos( θ ) + (1 − cos( θ )) (1 − cos( θ )) − sin( θ ) n x n y n z n y n z n x v y ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ n 2 (1 − cos( θ )) − sin( θ ) (1 − cos( θ )) + sin( θ ) cos( θ ) + (1 − cos( θ )) n x n z n y n y n z n x z

( n , θ ) R ( n , θ ) = I + sin( θ ) K + (1 − cos( θ )) K 2 n 2 ⎛ ⎞ cos( θ ) + (1 − cos( θ )) (1 − cos( θ )) − sin( θ ) (1 − cos( θ )) + sin( θ ) n x n y n z n x n z n y x ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ n 2 ⎜ ⎟ R ( n , θ ) = (1 − cos( θ )) + sin( θ ) cos( θ ) + (1 − cos( θ )) (1 − cos( θ )) − sin( θ ) n x n y n z n y n z n x y ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ n 2 (1 − cos( θ )) − sin( θ ) (1 − cos( θ )) + sin( θ ) cos( θ ) + (1 − cos( θ )) n x n z n y n y n z n x z

( , ) ( , ) n 1 θ 1 n 2 θ 2 ( , ) = ( , ) ∘ ( , ) n 3 θ 3 n 1 θ 1 n 2 θ 2 cos ( θ 3 ) = cos ( θ 1 ) cos ( θ 2 ) − sin ( θ 1 ) sin ( θ 2 ) ⋅ n 1 n 2 2 2 2 2 2 θ 2 θ 1 θ 1 θ 2 tan ( ) + tan ( ) − tan ( ) tan ( ) × n 2 n 1 n 1 n 2 1 2 2 2 2 = n 3 θ 3 θ 1 θ 2 tan ( ) 1 − tan ( ) tan ( ) ⋅ n 1 n 2 2 2 2

Im S 1 Re rotations sphere S 2 S 3 rotations sphere S 4

q = ( x , y , z , w ) q /∥ q ∥ q 1 q 2 = ( x , y , z , 0) q v q v q ⋆ = q q v ′

q = x i + y j + z k + w ( x , y , z ) w q = ( x , y , z , w ) ⎧ i 2 j 2 k 2 ⎪ = = = −1 ⎪ ⎪ ⎪ ⎪ ⎪ ij = − ji = k ⎨ jk = − kj = i ⎪ ⎪ ⎪ ⎪ ki = − ik = j ⎪ ⎩ ⎪ ijk = −1

q ⋆ = (− x , − y , − z , w ) − − − − − − − − − − − − − − −− − q q ⋆ x 2 y 2 z 2 w 2 ∥ q ∥ = = √ + + + ∥ q ∥ = 1 √ = ( i + j + k + ) ( i + j + k + ) = . . . q 1 q 2 x 1 y 1 z 1 w 1 x 2 y 2 z 2 w 2 ⎛ ⎞ + + − x 1 w 2 w 1 x 2 y 1 z 2 z 1 y 2 ⎜ ⎟ + + − y 1 w 2 w 1 y 2 z 1 x 2 x 1 z 2 ⎜ ⎟ = q 1 q 2 ( ) = ( ) = ⎜ ⎟ q 1 q 2 q 3 q 1 q 2 q 3 q 1 q 2 q 3 + + − z 1 w 2 w 1 z 2 x 1 y 2 y 1 x 2 ⎝ ⎠ ≠ q 1 q 2 q 2 q 1 − − − w 1 w 2 x 1 x 2 y 1 y 2 z 1 z 2 s = ( x , y , z ) w q = ( s , v ) = ( + + × , − ⋅ ) q 1 q 2 s 1 w 2 s 2 w 1 s 1 s 2 w 1 w 2 s 1 s 2

q = ( s , w ) ∥ q ∥ = 1 v = ( , , ) = ( v , 0) = ( , , , 0) v x v y v z q v v x v y v z q v q ⋆ v ′ x v ′ y v ′ = ( v ) = q = ( , , , 0) q v ′ R q q v ′ z v ′ v ′ x v ′ y v ′ = ( , , ) n = s /∥ s ∥ 2 acos( w ) v z w 2 s 2 ( v ) = ( s , w ) ( v , 0) (− s , w ) = . . . = (( − ) v + 2( s ⋅ v ) s + 2 w ( s × v ), 0) R q ∥ q ∥ = 1 q = ( s , w ) = ( n sin( ϕ ), cos( ϕ )) ∥ n ∥ = 1 cos 2 sin 2 sin 2 R q ( v ) = ( ( ( ϕ ) − ( ϕ )) v + 2 ( ϕ ) ( n ⋅ v ) n + 2 cos( ϕ ) sin( ϕ ) n × v , 0)                cos(2 ϕ ) 1−cos(2 ϕ ) sin(2 ϕ ) ⇒ 2 ϕ n q = ( n sin( θ /2), cos( θ /2)) θ n

( , ) ( , ) R 1 R 2 q 1 q 2 ∘ q 1 q 2 R 1 R 2 ( v ) = ∘ ( v ) R q 1 q 2 R q 1 R q 2 q 1 q 2 ) ⋆ ( v ) = ( ) v ( R q 1 q 2 q 1 q 2 q ⋆ 2 q ⋆ q 1 q 2 ) ⋆ q ⋆ 2 q ⋆ ( v ) = ( ) v ( ) ( = R q 1 q 2 q 1 q 2 1 1 q ⋆ 2 q ⋆ ( v ) = ( ) R q 1 q 2 q 1 q 2 v 1 q ⋆ ( v ) = ( v ) = ∘ ( v ) R q 1 q 2 q 1 R q 2 R q 1 R q 2 1

q = ( x , y , z , w ) y 2 z 2 ⎛ ⎞ 1 − 2( + ) 2( xy − wz ) 2( xz + wy ) ⎜ ⎟ x 2 z 2 R = 2( xy + wy ) 1 − 2( + ) 2( yz − wx ) ⎝ ⎠ x 2 y 2 2( xz − wy ) 2( yz + wx ) 1 − 2( + ) v ′ q v q ⋆ w 2 s 2 = ( v ) = q = (( − ) v + 2( s ⋅ v ) s + 2 w ( s × v ), 0) s = ( x , y , z ) R q T v ′ w 2 x 2 y 2 z 2 = ( − − − ) v + 2 ( ) v ( ) + 2 w ( ) × v x y z x y z x y z x 2 ⎛ ⎛ ⎞ ⎞ ⎛ ⎞ 0 − z xy xz y ⎜ w 2 ⎜ ⎟ ⎜ ⎟ ⎟ v ′ x 2 y 2 z 2 y 2 = ( − − − ) I + 2 + 2 w 0 − x v xy yz z ⎝ ⎠ ⎝ ⎝ ⎠ ⎠ z 2 − y 0 xz yz x v ′ x 2 y 2 z 2 w 2 = R v + + + = 1

v = ( , , ) = ( v , 0) = ( , , , 0) v x v y v z q v v x v y v z R 3 × 3 q = ( x , y , z , w ) ∥ q ∥ = 1 q q v q ⋆ Rv R 1 R 2 q 1 q 2 θ ⇒ q = ( n sin( θ /2), cos( θ /2)) n y 2 z 2 ⎛ ⎞ 1 − 2( + ) 2( xy − wz ) 2( xz + wy ) ⎜ ⎟ x 2 z 2 q = ( x , y , z , w ) ⇒ R = 2( xy + wy ) 1 − 2( + ) 2( yz − wx ) ⎝ ⎠ x 2 y 2 2( xz − wy ) 2( yz + wx ) 1 − 2( + )

q 1 q 2 t ∈ [0, 1] (1 − t ) + t q 1 q 2 q ( t ) = ∥(1 − t ) + t ∥ q 1 q 2

q 1 q 2 t ∈ [0, 1] sin((1 − t )Ω) sin( t Ω) q ( t ) = + cos(Ω) = ⋅ q 1 sin(Ω) q 2 q 1 q 2 sin(Ω) v 1 v 2 Ω v 1 v 2 θ = Ω t t ∈ [0, 1] v −cos(Ω) v 2 v 1 v ⊥ v ⊥ v = cos( θ ) + sin( θ ) = v 1 1 1 sin(Ω) sin(Ω) cos( θ )−cos(Ω) sin( θ ) sin( θ ) ⇒ v = ( ) + v 1 sin(Ω) v 2 sin(Ω) sin(Ω− θ ) sin( θ ) ⇒ v = + sin(Ω) v 1 sin(Ω)

q 1 q 2 t ∈ [0, 1] sin((1 − t )Ω) sin( t Ω) q ( t ) = + cos(Ω) = ⋅ q 1 sin(Ω) q 2 q 1 q 2 sin(Ω)

+ q − q n → − n θ → 2 π − θ ) − q − R ⇒ → − → ⋅ < 0 q 1 q 2 q 1 q 2 q 1 q 2 if ( dot(q1,q2)<0 ) q2 = -q2 q(t) = SLERP(q1,q2,t)