Quaternions

1. Quaternions • Hamilton conceived them as extended complex numbers • One real portion (s) and three imaginary portions (ai,bj,ck) • s + ai + bj + ck • s is a purely real scaler • a,b,c are real • i2 = j2 = k2 = -1 • ij = k = -ji • s2 + a2 + b2 + c2 = 1 • s=cos(θ/2), a=sin(θ/2)x, b=sin(θ/2)y, c=sin(θ/2)z • Where (x,y,z) is a unit vector along axis of rotation • Usually write quaternion as [v,w] with v=(a,b,c)

2. Why Quaternion Rotation? • Think of quaternion rotation as an enclosing sphere around an object. • By dragging the mouse along the sphere, the object is rotated • Ry(45)•Rx(90) not equal to Rx(90)•Ry(45) • Wouldn’t it be nice to rotate somewhere, and rotate right back? • Ry(180)•Rx(180) = Rz(180) • Now that’s confusing… • Instead of • glRotatef(θx, 1, 0, 0); • glRotatef(θy, 0, 1, 0); • glRotatef(θz, 0, 0, q); • Call once • Convert euler angle to quaternion • convert quaternion to axis angle • glRotatef(θ, x, y, z);

3. | 1-2b2-2c2 2ab-2sc 2ac+2sb 0 | | 2ab+2sc 1-2a2-2c2 2bc-2sa 0 | | 2ac -2sb 2bc+2sa 1-2a2 -2b2 0 | | 0 0 0 1| Book rewrites the matrix using a unit quaternion x,y,z. Therefore x2+y2 + z2=1 (The book renames it ux uy uz ) Recall s=cos(θ/2), a=sin(θ/2)x, b=sin(θ/2)y, c=sin(θ/2)z now s=cos(θ/2), a=sin(θ/2) ux, b=sin(θ/2) uy, c=sin(θ/2) uz I will convert M11 of 5-107 to M11 of 5-108 Using geometric identities sin2(θ/2) = ½(1-cos θ), cos2(θ/2) = ½(1+cos θ) , 1-2 sin2(θ/2)=cos θ 1-2b2-2c2 =1-2 sin2(θ/2)y2- 2 sin2(θ/2) z2 = 1-2 sin2(θ/2)(y2 +z2) = 1-2 sin2(θ/2)(1-x2) = 1-2 sin2(θ/2) - 2 sin2(θ/2) x2 = cos θ – (2)½(1-cos θ) x2 = x2 (1-cos θ) + cos θ = ux (1-cos θ) + cos θ Quaternion Matrix