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Orientation

Orientation. Rotation about principle axes - fixed angles. Rotation about object’s axes - Euler angles. Axis-angle rotation. Quaternion. Orientation. 1 0 0 0 cos( q ) -sin( q ) 0 sin( q ) cos( q ). We know how to rotate about the global axes.

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Orientation

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  1. Orientation Rotation about principle axes - fixed angles Rotation about object’s axes - Euler angles Axis-angle rotation Quaternion

  2. Orientation 1 0 0 0 cos(q) -sin(q) 0 sin(q) cos(q) We know how to rotate about the global axes cos(q) 0 sin(q) 0 1 0 -sin(q) 0 cos(q) cos(q) -sin(q) 0 sin(q) cos(q) 0 0 0 1

  3. y x z Fixed angles Rotate about global axes in a fixed order Rotating about global axes is what the rotation matrices do Can use most any triple of axes Rotate about x, then y, then z (10, 90, -45)

  4. y x z Gimbal lock From some orientations, can’t do some rotations (0,90,0) Can’t rotate around x-axis

  5. y x z Euler angles Rotate about axes of object Can use most any triple of axes Roll, Pitch, Yaw (10, 90, -45)

  6. Equivalence of Fixed angles and Euler angles Ru(a) = Rx(a) Rv (b)Ru (a) = Rx(a)Ry (b)Rx (- a)Rx (a) = Rx (a)Ry (b) Rw (g) Rv (b)Ru (a) = Rx (a)Ry (b) Rz (g)

  7. Quaternions Keep axis-angle orientation as 4-tuple Q = (s, x, y, z) = (s,v) Q1*Q2 = (s1,v1)*(s2,v2) = (s1*s2+v1*v2 , s1*v2 + s2*v1 + v1xv2) Q1+Q2 = (s1,v1)+(s2,v2) = (s1+s2, v1+v2)

  8. Quaternions Keep axis-angle orientation as 4-tuple (sin(t/2), cos(t/2)*(x,y,z))

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