Orientation Rotation about principle axes fixed angles Rotation
Orientation Rotation about principle axes - fixed angles Rotation about object’s axes - Euler angles Axis-angle rotation Quaternion Rick Parent - CIS 681
Orientation We know how to rotate about the global axes cos(q) 0 sin(q) 0 1 0 -sin(q) 0 cos(q) 1 0 0 0 cos(q) -sin(q) 0 sin(q) cos(q) 0 0 0 1 Rick Parent - CIS 681
Fixed angles Rotate about global axes in a fixed order Rotating about global axes is what the rotation matrices do y Can use most any triple of axes Rotate about x, then y, then z x (10, 90, -45) z Rick Parent - CIS 681
Gimbal lock From some orientations, can’t do some rotations y (0, 90, 0) x Can’t rotate around x-axis z Rick Parent - CIS 681
Euler angles Rotate about axes of object Can use most any triple of axes y Roll, Pitch, Yaw (10, 90, -45) x z Rick Parent - CIS 681
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) Rick Parent - CIS 681
Quaternions Keep axis-angle orientation as 4 -tuple Q = (s, x, y, z) = (s, v) Q 1*Q 2 = (s 1, v 1)*(s 2, v 2) = (s 1*s 2+v 1*v 2 , s 1*v 2 + s 2*v 1 + v 1 xv 2) Q 1+Q 2 = (s 1, v 1)+(s 2, v 2) = (s 1+s 2, v 1+v 2) Rick Parent - CIS 681
Quaternions Keep axis-angle orientation as 4 -tuple (sin(t/2), cos(t/2)*(x, y, z)) Rick Parent - CIS 681
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