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q̂ = qx,qy,qz,qw( ), qv = qx,qy,qz( )⇒ q̂ = qv,qw( )

q̂ = qv,qw( ) = iqx + jqy + kqz + qw = qv + qwqv = iqx + jqy + kqz( ) = qx,qy,qz( )i2 = j2 = k2 = ijk = −1jk = −kj = i, ki = −ik = j, ij = − ji = k

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q̂r̂ = iqx + jqy + kqz + qw( ) irx + jry + krz + rw( )= i qyrz − qzry + rwqx + qwrx( )+ j qzrx − qxrz + rwqy + qwry( )+k qxry − qyrx + rwqz + qwrz( )+qwrw − qxrx − qyry − qzrz= qv × rv + rwqv + qwrv,qwrw −qv ⋅ rv( )

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/* !** p <- q * r !*/ !void qmul(float *p, const float *q, const float *r) !{ ! p[0] = q[1]*r[2] - q[2]*r[1] + r[3]*q[0] + q[3]*r[0]; ! p[1] = q[2]*r[0] - q[0]*r[2] + r[3]*q[1] + q[3]*r[1]; ! p[2] = q[0]*r[1] - q[1]*r[0] + r[3]*q[2] + q[3]*r[2]; ! p[3] = q[3]*r[3] - q[0]*r[0] - r[1]*q[1] - q[2]*r[2]; !} !

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q̂+ r̂ = qv,qw( )+ rv, rw( ) = qv + rv,qw + rw( )

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( )1,ˆ 0i =

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=

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( ) ( ) φφφφ cossincos,sinˆ1ˆ +==⇒= qqn uuqq

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( ) ( ) qqe uq u φφ == logˆlog

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qt q φφφφ φ cossincossinˆ +==+= uuq u

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=

ppp

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✓

2

, 0, 0, cos✓

2

◆

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2

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✓

2

, 0, cos✓

2

◆

= j sin✓

2

+ cos

✓

2

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✓0, 0, sin

✓

2

, cos✓

2

◆

= k sin✓

2

+ cos

✓

2

�¦-�x�±&�,�H¶Q®³�/* !** q <- �(x, y, z) x8(a) !*/ !void qmake(float *q, float x, float y, float z, float a) !{ ! float l = x * x + y * y + z * z; !! if (l != 0.0f) { ! float s = sin(a *= 0.5f) / sqrt(l); !! q[0] = x * s; ! q[1] = y * s; ! q[2] = z * s; ! q[3] = cos(a); ! } !} !

14

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ˆq = iqx

+ jqy

+ kqz

+ qw

= sin ✓ (iux

+ iuy

+ iuz

) + cos ✓

cos

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q̂ = (qx

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)

s = 2/n(q̂)

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m[ 0] = 1.0f - yy - zz;! m[ 1] = xy + zw;! m[ 2] = zx - yw;! m[ 4] = xy - zw;! m[ 5] = 1.0f - zz - xx;! m[ 6] = yz + xw;! m[ 8] = zx + yw;! m[ 9] = yz - xw;! m[10] = 1.0f - xx - yy;! m[ 3] = m[ 7] = m[11] =! m[12] = m[13] = m[14] = 0.0f;! m[15] = 1.0f;!}!

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tr Mq( ) = miiq

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∑ =m00q +m11

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q = 4− 4 qx2 + qy

2 + qz2( ) = 4qw2

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qw =12tr Mq( )

4qx2 = +m00

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4qz2 = −m00

q −m11q +m22

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qx =m21

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t

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R s, t( ) =

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0 0 0 1

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u = ux uy uz( ) = s× ts× t

e = cos2φ = s ⋅ t

h = 1− cos2φsin2 2φ

=1− eu ⋅u

q̂ = qv,qw( ) = 12 1+ e( )

s× t,2 1+ e( )2

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const double cycle(5.0); !… !// k�J�©ÔÅÇÉ !glfwSetTime(0.0); !… !// º¸×ʺ���¤�³��²��F\�³�while (window.shouldClose() == GL_FALSE) !{ !��… !��// J�©zTÙ*LÞ_Ú!��float t((float)(fmod(glfwGetTime(), cycle) / cycle)); !��… !

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P t( ) = P0 1− t( )+P1t

P0

P1

P(t)

t

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/* !** 7s`#0Gs�¶Q®³ !*/ !void translate(float *m, float x, float y, float z) !{ ! m[ 3] = x; m[ 7] = y; m[11] = z; ! m[ 0] = m[ 5] = m[10] = m[15] = 1.0f; ! m[ 1] = m[ 2] = m[ 4] = ! m[ 6] = m[ 8] = m[ 9] = ! m[12] = m[13] = m[14] = 0.0f; !} !

27

2V�¶]m`#�³0Gs�¶>³�/* !** 7s`#·ËÑØÂÒש0Gs�¶Q®³ !*/ !void linearMotion( ! float *m, const float *p0, const float *p1, float t) !{ ! float x = (p1[0] - p0[0]) * t + p0[0]; ! float y = (p1[1] - p0[1]) * t + p0[1]; ! float z = (p1[2] - p0[2]) * t + p0[2]; ! translate(m, x, y, z); !} !

28

Cubic Hermite Spline�

P t( ) = P0 2t3 −3t2 +1( )+m0 t3 − 2t2 + t( )

+P1 −2t3 +3t2( )+m1 t

3 − t2( )P0

P1P(t)

tm0

m1

29

Catmull-Rom Spline�• Cubic Hermite Spline ¨��¤�

m0 =P1 −P−12

m1 =P2 −P02

30

P0

P1P(t)

t m0 m1

P-1P2

Catmull-Rom Spline ©Á×ÎÕÀØÊ�/* !** Catmull-Rom Spline !*/�static float catmull_rom( ! float x0, float x1, float x2, float x3, float t) !{ ! float m0 = (x2 - x0) * 0.5f, m1 = (x3 - x1) * 0.5f; !! return (((2.0f * x1 - 2.0f * x2 + m0 + m1) * t ! - 3.0f * x1 + 3.0f * x2 - 2.0f * m0 - m1) * t ! + m0) * t + x1; !} !

31

Catmull-Rom Spline ¨°³u��/* !** Catmull-Rom Spline ¨°³V�©u�!*/�void interpolate( ! float *p, ! const float *p0, const float *p1, ! const float *p2, const float *p3, ! float t) !{ ! p[0] = catmull_rom(p0[0], p1[0], p2[0], p3[0], t); ! p[1] = catmull_rom(p0[1], p1[1], p2[1], p3[1], t); ! p[2] = catmull_rom(p0[2], p1[2], p2[2], p3[2], t); !} !

32

Kochanek-Bartels Spline�• Cubic Hermite Spline ¨��¤�

si =1− t( ) 1+ b( ) 1− c( )

2Pi −Pi−1( )+

1− t( ) 1− b( ) 1+ c( )2

Pi+1 −Pi( )

di =1− t( ) 1+ b( ) 1+ c( )

2Pi −Pi−1( )+

1− t( ) 1− b( ) 1− c( )2

Pi+1 −Pi( )

Pisi di+1Pi+1

Pi-1

m0 m1

si+1di

Pi+2

t: Tension b: Bias c: Continuity�

33

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q̂t = 1− t( ) q̂0 + tq̂n, t ∈ 0,1[ ]q̂t=0 = q̂0q̂t+Δt = 1− t +Δt( )( ) q̂0 + t +Δt( ) q̂n

= 1− t( ) q̂0 + tq̂n +Δt q̂n − q̂0( )= q̂t +Δq̂ qn� q0�qt�qt+�t�

Δq̂ = Δt q̂n − q̂0( )��¥�

34

Z�m<u��(Slerp) ( ) ( ) [ ]1,0,ˆˆˆ,ˆ,ˆˆ 1 ∈= − tt tqqrrqs

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qrqsˆ,ˆ,ˆˆ1ˆ,ˆ,ˆˆ0

→⇒→

→⇒→

tttt

( ) ( ) ( )( ) ( )rqrqrqs ˆsinsinˆ

sin1sin,ˆ,ˆ,ˆ,ˆˆ

φφ

φφ tttt +

−==slerp

( )

( )21

ˆˆ1sin

ˆˆcos

ˆˆcos

rq

rq

rq

⋅−=

⋅=

+++=⋅=−

φ

φ

φ wwzzyyxx rqrqrqrq

q̂

r̂

( )t,ˆ,ˆˆ rqs

35

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#include <cmath> !!/* !** p ← q ¦�r ¶ t ¥u��*/ !void slerp(float *p, !����������const float *q, !����������const float *r, !����������const float t) !{ !��float�qr #= q[0] * r[0] !

#+ q[1] * r[1] !#+ q[2] * r[2] !#+ q[3] * r[3]; !

float ss #= 1.0f - qr * qr; !!

if (ss == 0.0) { ! p[0] = q[0]; ! p[1] = q[1]; ! p[2] = q[2]; ! p[3] = q[3]; ! } !��else { !����float sp = sqrt(ss); !����float ph = acos(qr); !����float pt = ph * t; !����float t1 = sin(pt) / sp; ! ���float t0 = sin(ph - pt)�/ sp; ! !����p[0] = q[0] * t0 + r[0] * t1; !����p[1] = q[1] * t0 + r[1] * t1; !����p[2] = q[2] * t0 + r[2] * t1; !����p[3] = q[3] * t0 + r[3] * t1; ! } !} !!

36

2��©,�H©u��

( )itii −= + ,ˆ,ˆˆ 1qqq Slerp[ ]1ˆ,ˆ +ii qq%�� ¨��¤�

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11

11

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iq̂

1ˆ −iq

1ˆ +iq

38

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