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ゲームグラフィックス特論 第4回 講義ノート
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ゲームグラフィックス特論 第4回 講義ノート

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四元数ほか。まだ書きかけです。

四元数ほか。まだ書きかけです。

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  • 1. 119 8 8.1 ( ) quaternion ( ) a, b i a + bi w, x, y, z i, j, k w + xi + yj + zk ! qx, qy, qz, qw (24) (qx, qy, qz) qv (25) ! (26) 8.2 ! (27) qv (28) ! qv qw (29) ˆq = (qx, qy, qz, qw) qv = (qx, qy, qz) ˆq = (qv, qw) ˆq = (qv, qw) = iqx + jqy + kqz + qw qv = (qx, qy, qz) = (iqx + jqy + kqz) ˆq = qv + qw
  • 2. 120 i, j, k (30) 8.3 8.3.1 (31) C qmul() p, q, r x, y, z, w 4 q r p /** ***p*<K*q***r* */* void*qmul(float**p,*const*float**q,*const*float**r)* {* **p[0]*=*q[1]*r[2]*K*q[2]*r[1]*+*r[3]*q[0]*+*q[3]*r[0];* **p[1]*=*q[2]*r[0]*K*q[0]*r[2]*+*r[3]*q[1]*+*q[3]*r[1];* **p[2]*=*q[0]*r[1]*K*q[1]*r[0]*+*r[3]*q[2]*+*q[3]*r[2];* **p[3]*=*q[3]*r[3]*K*q[0]*r[0]*K*r[1]*q[1]*K*q[2]*r[2];* }* 8.3.2 (32) 8.3.3 (33) 8.3.4 (34) i2 = j2 = k2 = ijk = 1 jk = kj = i, ki = ik = j, ij = ji = k ˆqˆr = (iqx + jqy + kqz + qw)(irx + jry + krz + rw) = i(qyrz qzry + rwqx + qwrx) + j(qzrx qxrz + rwqy + qwry) + j(qxry qyrx + rwqz + qwrz) + qwrw qxrx qyry qzrz = (qv ⇥ rv + rwqv + qwrv, qwrw qv · rv) ˆq + ˆr = (qv, qw) + (rv, rw) = (qv + rv, qw + rw) ˆq⇤ = (qv, qw)⇤ = ( qv, qw) n(ˆq) = p ˆqˆq⇤ = p ˆq⇤ˆq = q q2 x + q2 y + q2 z + q2 w
  • 3. 121 8.3.5 (35) (31) ! = ! (36) 8.3.6 (37) (38) 8.4 8.4.1 (39) (40) (41) 8.4.2 (42) (43) ˆi = (0, 1) ˆqˆi = (qv ⇥ rv + rwqv + qwrv, qwrw qv · rv) = (qv ⇥ 0 + 1qv + qw0, qw1 qv · 0) = (qv, qw) = ˆq n(ˆq) = p ˆqˆq⇤ , p ˆqˆq⇤ n(ˆq) = 1 ˆq 1 = ˆq⇤ |n(ˆq)|2 (ˆq⇤ )⇤ = ˆq (ˆq + ˆr)⇤ = ˆq⇤ + ˆr⇤ (ˆqˆr)⇤ = ˆr⇤ ˆq⇤ n(ˆq⇤ ) = n(ˆq) n(ˆqˆr) = n(ˆq)n(ˆr)
  • 4. 122 8.4.3 (44) (45) 8.5 8.5.1 ( ) 1 (46) ! !! !! = 1 (47) ! 1 (48) cos ! + ! sin ! = !!" ! (49) 8.5.2 (49) !! (50) 8.5.3 (49) (51) 8.5.4 (38) ! ! = 1 (52) ˆp(sˆq + tˆr) = sˆpˆq + tˆpˆr ˆp(ˆqˆr) = (ˆpˆq)ˆr n(ˆq) = 1 ˆq = (sin uq, cos ) = sin uq + cos n(ˆq) = n(sin uq, cos ) = q sin2 (uq · uq) + cos2 = q sin2 + cos2 = 1 ˆq = sin uq + cos = e uq log ˆq = log(e uq ) = uq ˆqt = (sin uq + cos )t = e tuq = sin t uq + cos t ˆq 1 = ˆq⇤ |n(ˆq)|2 = ˆq⇤
  • 5. 123 8.5.5 p (53) ! pv = (px, py, pz) (54) ! (55) !′ 100 ! !! 2! (56) !!! = !∗ (57) 100 ! −! (58) " X (59) p = 0 B B @ px py pz pw 1 C C A ˆp = (pv, pw) pv = (px, py, pz) ˆq = (sin uq, cos ) = sin uq + cos ˆp0 = ˆqˆpˆq 1 ˆp0 = ˆqˆpˆq⇤ uq φ" φ" ˆp ˆp´ ˆp0 = ( ˆq)ˆp( ˆq⇤ ) = ˆqˆpˆq⇤ ˆqx(✓) = ✓ sin ✓ 2 , 0, 0, cos ✓ 2 ◆ = i sin ✓ 2 + cos ✓ 2
  • 6. 124 " Y (60) " Z (61) " C qmake() x, y, z a q /** ***q*<K* (x,*y,*z)* (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);* **}* }* 8.5.6 ! ! (62) ! ! !! ˆqy(✓) = ✓ 0, sin ✓ 2 , 0, cos ✓ 2 ◆ = j sin ✓ 2 + cos ✓ 2 ˆqz(✓) = ✓ 0, 0, sin ✓ 2 , cos ✓ 2 ◆ = k sin ✓ 2 + cos ✓ 2 ˆr(ˆqˆpˆq⇤ )ˆr⇤ = (ˆrˆq)ˆp(ˆq⇤ ˆr⇤ ) = (ˆrˆq)ˆp(ˆrˆq)⇤
  • 7. 125 8.5.7 ! = (!!, !!, !!, !!) (63) ! = 2/!(!) ! (64) " C qrot() q m /** *** *m*<K* *q* */* void*qrot(float**m,*const*float**q)* {* **float*xx*=*q[0]***q[0]***2.0f;* **float*yy*=*q[1]***q[1]***2.0f;* **float*zz*=*q[2]***q[2]***2.0f;* **float*xy*=*q[0]***q[1]***2.0f;* **float*yz*=*q[1]***q[2]***2.0f;* **float*zx*=*q[2]***q[0]***2.0f;* **float*xw*=*q[0]***q[3]***2.0f;* **float*yw*=*q[1]***q[3]***2.0f;* **float*zw*=*q[2]***q[3]***2.0f;* * **m[*0]*=*1.0f*K*yy*K*zz;* **m[*1]*=*xy*+*zw;* **m[*2]*=*zx*K*yw;* **m[*4]*=*xy*K*zw;* **m[*5]*=*1.0f*K*zz*K*xx;* **m[*6]*=*yz*+*xw;* **m[*8]*=*zx*+*yw;* **m[*9]*=*yz*K*xw;* **m[10]*=*1.0f*K*xx*K*yy;* **m[*3]*=*m[*7]*=*m[11]*=* **m[12]*=*m[13]*=*m[14]*=*0.0f;* **m[15]*=*1.0f;* }* Mq = 0 B B @ 1 s(q2 y + q2 z) s(qxqy qwqz) s(qzqx + qwqy) 0 s(qxqy + qwqz) 1 s(q2 z + q2 x) s(qyqz qwqx) 0 s(qzqx qwqy) s(qyqz + qwqx) 1 s(q2 x + q2 y) 0 0 0 0 1 1 C C A Mq = 0 B B @ 1 2(q2 y + q2 z) 2(qxqy qwqz) 2(qzqx + qwqy) 0 2(qxqy + qwqz) 1 2(q2 z + q2 x) 2(qyqz qwqx) 0 2(qzqx qwqy) 2(qyqz + qwqx) 1 2(q2 x + q2 y) 0 0 0 0 1 1 C C A
  • 8. 126 8.5.8 Mq (65) ( ) (66) qw (67) qx, qy, qz (68) (69) (70) qx, qy, qz (71) 8.6 s t s t u 2φ Mq = 0 B B @ mq 00 mq 01 mq 02 mq 03 mq 10 mq 11 mq 12 mq 13 mq 20 mq 21 mq 22 mq 23 mq 30 mq 31 mq 32 mq 33 1 C C A = 0 B B @ 1 2(q2 y + q2 z) 2(qxqy qwqz) 2(qzqx + qwqy) 0 2(qxqy + qwqz) 1 2(q2 z + q2 x) 2(qyqz qwqx) 0 2(qzqx qwqy) 2(qyqz + qwqx) 1 2(q2 x + q2 y) 0 0 0 0 1 1 C C A tr(Mq ) = 3X i=0 mq ii = m00 + m11 + m22 + m33 = 4 4(q2 x + q2 y + q2 z) = 4q2 w qw = 1 2 p tr(Mq) qx = mq 21 mq 12 4qw qy = mq 02 mq 20 4qw qz = mq 10 mq 01 4qw qx = 1 2 q +mq 00 mq 11 mq 22 + mq 33 qy = 1 2 q mq 00 + mq 11 mq 22 + mq 33 qz = 1 2 q mq 00 mq 11 + mq 22 + mq 33
  • 9. 127 101 ! u (102) (103) !×! = sin 2! = 2 sin ! cos ! (102) (104) ! ∙ ! = cos 2! = 2 cos! ! − 1 = ! (105) (106) R(s, t) (107) s t 2φ u ˆq = (sin u, cos ) u = s ⇥ t ks ⇥ tk ˆq = ✓ sin sin 2 s ⇥ t, cos ◆ = ✓ 1 2 cos s ⇥ t, cos ◆ cos = r 1 + e 2 ˆq = 1 p 2(1 + e) s ⇥ t, p 2(1 + e) 2 ! R(s, t) = 0 B B @ e + hu2 x huxuy uz huzux + uy 0 huxuy + uz e + hu2 y huyuz ux 0 huzux uy huyuz + ux e + hu2 z 0 0 0 0 1 1 C C A u = (ux, uy, uz) = s ⇥ t ks ⇥ tk e = cos 2 = s · t h = 1 cos 2 sin2 2 = 1 e u · u