( , ), 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] pi ti p(t), t ∈ [ , ] t0 tN−1 p( ) = ti pi

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

p(t) = ( ) ∑

i=0 d

∑

j=0 N−1

cj

i pj

ti p(t) = (t) ∑

i=0 N−1

αi pi ( ) αi d

∀t ∈ [ , ] , p(t) = (t) t0 tN−1 ∑

i=0 N−1

αi pi ∀i ∈ [0, N − 1] p( ) = ti pi N − 1 p(t) = (t) ∑

i=0 N−1

αi pi (t) = αi ∏

k=0,k≠i N−1

t − tk − ti tk ( ) = 1 αi ti ( ) = 0 αi tk

C 2

(p0,t0) (p1,t1) (p2,t2) (p3,t3) (p4,t4) (p5,t5)

{ p(s) = + + s + c3s3 c2s2 c1 c0 p(0) = , p(1) = , (0) = , (1) = p0 p1 p′ d0 p′ d1 ⇒ ⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + c3 3 + c3 + c2 2 + c2 + c1 c1 c1 c0 c0 = p0 = p1 = d0 = d1 ⇒ ⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = c0 p0 = c1 d0 = −3 + 3 − 2 − c2 p0 p1 d0 d1 = 2 − 2 + + c3 p0 p1 d0 d1 ∀s ∈ [0, 1] , p(s) = (2 − 3 + 1) + ( − 2 + s) + (−2 + 3 ) + ( − ) s3 s2 p0 s3 s2 d0 s3 s2 p1 s3 s2 d1 t ∈ [ , ] ti ti+1 s = t − ti − ti+1 ti d1 p0 p1 d0

= μ di − pi+1 pi−1 − ti+1 ti−1 μ ∈ [0, 2] μ = 1

p(t) ( , pi ti)i∈[0,N−1] t ∈ [ , ] t1 tN−2 i t ∈ [ , ] ti ti+1 = μ di − pi+1 pi−1 − ti+1 ti−1 = μ di+1 − pi+2 pi − ti+2 ti p(t) = (2 − 3 + 1) + ( − 2 + s) + (−2 + 3 ) + ( − ) s3 s2 pi s3 s2 di s3 s2 pi+1 s3 s2 di+1 s = t − ti − ti+1 ti

C 1 C 2 C 2

= pi ∑k ωk bi

k

p = B ω p = + ( ) b0 ∑k − bk b0     

dk

p = + D ω b0 ⇒ D ⇒

t w t w t w

(x, y, z) ( , , ) sx sy sz ( , , , ) qx qy qz qw

w p(t) = f(w(t))

→ = R1 z1 ei θ1 → = R2 z2 ei θ2 → z = Rα

1→2

ei (α

+(1−α) ) θ1 θ2

⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ R = ⎛ ⎝ ⎜ r00 r10 r20 r01 r11 r21 r02 r12 r22 ⎞ ⎠ ⎟ R = Id RT = Rv v′ R = R1 R2

M = (1 − α) + α R1 R2 M

(x, y, z)

= Rx ⎛ ⎝ ⎜ cos(θ) − sin(θ) sin(θ) cos(θ) 1 ⎞ ⎠ ⎟ = Ry ⎛ ⎝ ⎜ cos(θ) − sin(θ) 1 sin(θ) cos(θ) ⎞ ⎠ ⎟ = Rz ⎛ ⎝ ⎜ 1 cos(θ) − sin θ) sin(θ) cos(θ) ⎞ ⎠ ⎟

n θ

u1 u2 n = ( × )/∥ × ∥ u1 u2 u1 u2 θ = acos( ⋅ ) u1 u2 u2

(n, θ) v v = + v∥ v⊥ = + v′ v′

∥

v′

⊥

⇒ = + (cos(θ) + sin(θ) n × ) v′ v∥ v⊥ v⊥ = (v ⋅ n) n v∥ = v − (v ⋅ n) n 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 v′ = cos(θ)v + sin(θ) × v + (1 − cos(θ)) ( ) v v′ ⎛ ⎝ ⎜ nx ny nz ⎞ ⎠ ⎟ nx ny nz ⎛ ⎝ ⎜ nx ny nz ⎞ ⎠ ⎟ = cos(θ) v + sin(θ) v + (1 − cos(θ)) v v′ ⎛ ⎝ ⎜ nz −ny −nz nx ny −nx ⎞ ⎠ ⎟     

K

⎛ ⎝ ⎜ n2

x

nx ny nx nz nx ny n2

y

ny nz nx nz ny nz n2

z

⎞ ⎠ ⎟     

K2

= v v′ ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ cos(θ) + (1 − cos(θ)) n2

x

(1 − cos(θ)) + sin(θ) nxny nz (1 − cos(θ)) − sin(θ) nxnz ny (1 − cos(θ)) − sin(θ) nxny nz cos(θ) + (1 − cos(θ)) n2

y

(1 − cos(θ)) + sin(θ) nynz nx (1 − cos(θ)) + sin(θ) nxnz ny (1 − cos(θ)) − sin(θ) nynz nx cos(θ) + (1 − cos(θ)) n2

z

⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

(n, θ) R(n, θ) = I + sin(θ) K + (1 − cos(θ)) K2 R(n, θ) = ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ cos(θ) + (1 − cos(θ)) n2

x

(1 − cos(θ)) + sin(θ) nxny nz (1 − cos(θ)) − sin(θ) nxnz ny (1 − cos(θ)) − sin(θ) nxny nz cos(θ) + (1 − cos(θ)) n2

y

(1 − cos(θ)) + sin(θ) nynz nx (1 − cos(θ)) + sin(θ) nxnz ny (1 − cos(θ)) − sin(θ) nynz nx cos(θ) + (1 − cos(θ)) n2

z

⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

( , ) n1 θ1 ( , ) n2 θ2 ( , ) = ( , ) ∘ ( , ) n3 θ3 n1 θ1 n2 θ2 cos( ) = cos( ) cos( ) − sin( ) sin( ) ⋅

θ3 2 θ1 2 θ2 2 θ1 2 θ2 2

n1 n2 = n3 1 tan( )

θ3 2

tan( ) + tan( ) − tan( ) tan( ) ×

θ2 2

n2

θ1 2

n1

θ1 2 θ2 2

n1 n2 1 − tan( ) tan( ) ⋅

θ1 2 θ2 2

n1 n2

S1 S3

Re

Im

rotations sphere S2

sphere S4 rotations

q = (x, y, z, w) q/∥q∥ q1 q2 = (x, y, z, 0) qv = q qv′ qv q⋆

q = x i + y j + z k + w w (x, y, z) q = (x, y, z, w) ⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = = = −1 i2 j2 k2 ij = −ji = k jk = −kj = i ki = −ik = j ijk = −1

= (−x, −y, −z, w) q⋆ ∥q∥ = = q q⋆ −− − √ + + + x2 y2 z2 w2 − − − − − − − − − − − − − − √ ∥q∥ = 1 = ( i + j + k + ) ( i + j + k + ) = . . . q1 q2 x1 y1 z1 w1 x2 y2 z2 w2 = q1 q2 ⎛ ⎝ ⎜ ⎜ ⎜ + + − x1w2 w1x2 y1z2 z1y2 + + − y1w2 w1y2 z1x2 x1z2 + + − z1w2 w1z2 x1y2 y1x2 − − − w1w2 x1x2 y1y2 z1z2 ⎞ ⎠ ⎟ ⎟ ⎟ w s = (x, y, z) q = (s, v) = ( + + × , − ⋅ ) q1 q2 s1 w2 s2 w1 s1 s2 w1 w2 s1 s2 ( ) = ( ) = q1q2 q3 q1 q2q3 q1 q2 q3 ≠ q1 q2 q2 q1

q = (s, w) ∥q∥ = 1 v = ( , , ) vx vy vz = (v, 0) = ( , , , 0) qv vx vy vz = (v) = q qv′ Rq qv q⋆ = ( , , , 0) qv′ v′

x v′ y v′ z

= ( , , ) v′ v′

x v′ y v′ z

v n = s/∥s∥ 2 acos(w)

(v) = (s, w) (v, 0) (−s, w) = . . . = (( − ) v + 2(s ⋅ v) s + 2w (s × v), 0) Rq w2 s2 ∥q∥ = 1 q = (s, w) = (n sin(ϕ), cos(ϕ)) ∥n∥ = 1 (v) = ( v + (n ⋅ v) n + n × v , 0) Rq ( (ϕ) − (ϕ)) cos2 sin2     

cos(2ϕ)

2 (ϕ) sin2     

1−cos(2ϕ)

2 cos(ϕ) sin(ϕ)     

sin(2ϕ)

⇒ n 2ϕ

q = (n sin(θ/2), cos(θ/2)) θ n

( , ) R1 R2 ( , ) q1 q2 q1 q2 ∘ R1 R2 (v) = ∘ (v) Rq1 q2 Rq1 Rq2 (v) = ( ) v ( Rq1 q2 q1q2 q1q2)⋆ (v) = ( ) v ( ) Rq1 q2 q1 q2 q⋆

2 q⋆ 1

( = q1q2)⋆ q⋆

2 q⋆ 1

(v) = ( v ) Rq1 q2 q1 q2 q⋆

2 q⋆ 1

(v) = (v) = ∘ (v) Rq1 q2 q1 Rq2 q⋆

1

Rq1 Rq2

q = (x, y, z, w) R = ⎛ ⎝ ⎜ 1 − 2( + ) y2 z2 2(xy + wy) 2(xz − wy) 2(xy − wz) 1 − 2( + ) x2 z2 2(yz + wx) 2(xz + wy) 2(yz − wx) 1 − 2( + ) x2 y2 ⎞ ⎠ ⎟ = (v) = q = (( − ) v + 2(s ⋅ v) s + 2w (s × v), 0) v′ Rq qv q⋆ w2 s2 s = (x, y, z) = ( − − − )v + 2 ( ) v + 2w ( ) × v v′ w2 x2 y2 z2 x y z ( ) x y z

T

x y z = ( − − − ) I + 2 + 2w v v′ ⎛ ⎝ ⎜ w2 x2 y2 z2 ⎛ ⎝ ⎜ x2 xy xz xy y2 yz xz yz z2 ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ z −y −z x y −x ⎞ ⎠ ⎟ ⎞ ⎠ ⎟ = R v v′ + + + = 1 x2 y2 z2 w2

v = ( , , ) vx vy vz = (v, 0) = ( , , , 0) qv vx vy vz R 3 × 3 q = (x, y, z, w) ∥q∥ = 1 Rv q qv q⋆ R1 R2 q1 q2 n θ ⇒ q = (n sin(θ/2), cos(θ/2)) q = (x, y, z, w) ⇒ R = ⎛ ⎝ ⎜ 1 − 2( + ) y2 z2 2(xy + wy) 2(xz − wy) 2(xy − wz) 1 − 2( + ) x2 z2 2(yz + wx) 2(xz + wy) 2(yz − wx) 1 − 2( + ) x2 y2 ⎞ ⎠ ⎟

q1 q2 t ∈ [0, 1] q(t) = (1 − t) + t q1 q2 ∥(1 − t) + t ∥ q1 q2

q1 q2 t ∈ [0, 1] q(t) = + sin((1 − t)Ω) sin(Ω) q1 sin(tΩ) sin(Ω) q2 cos(Ω) = ⋅ q1 q2

v1 v2 Ω v1 v2 v θ = Ωt t ∈ [0, 1] v = cos(θ) + sin(θ) v1 v⊥

1

= v⊥

1 −cos(Ω) v2 v1 sin(Ω)

⇒ v = ( ) +

sin(Ω) cos(θ)−cos(Ω) sin(θ) sin(Ω)

v1

sin(θ) sin(Ω) v2

⇒ v = +

sin(Ω−θ) sin(Ω) v1 sin(θ) sin(Ω)

q1 q2 t ∈ [0, 1] q(t) = + sin((1 − t)Ω) sin(Ω) q1 sin(tΩ) sin(Ω) q2 cos(Ω) = ⋅ q1 q2

if( dot(q1,q2)<0 ) q2 = -q2 q(t) = SLERP(q1,q2,t)

+q −q n → −n θ → 2π − θ) −q −R ⇒ → − q1 q2 → q1 q2 ⋅ < 0 q1 q2

r p ⇒ ( , ) p1 r1 ( , ) p2 r2 t ∈ [0, 1] p(t) = (1 − t) + t p1 p2 ( , ) → ( , ) r1 r2 q1 q2 q(t) = SLERP( , , t) q1 q2 q(t) → r(t)

value value Close Controls Close Controls

M ⇒ M ⇒ M = R D R D SVD(M) = WΣ V T R = WVT D = VΣVT R = M , = 0.5 ( + ) R0 Ri+1 Ri R−T

i

4 × 4 , M1 M2 , p1 p2 , M1 M2 R1 R2 3 × 3 D1 D2 3 × 3 , M1 M2 p(t) = (1 − t) + t p1 p2 D(t) = (1 − t) + t D1 D2 q1 q2 R1 R2 M → q q = ( , , , )

− Mzy Myz 2r − Mxz Mzx 2r − Myx Mxy 2r r 2

r = 1 + + + Mxx Myy Mzz − − − − − − − − − − − − − − − − − √ q(t) = SLERP( , , t) q1 q2 q(t) → R(t) M(t) = R(t) D(t) p(t)