The plane of rotation r equals cosine theta multiplied by the elevation plus sine theta multiplied by the axis a crossed with projection p. The point where the elevation (lime green) joins the...v' = ∑ wi (v) (Qi * v + ti) where Qi is the absolute bone (or joint) rotation (e.g. stored as a quaternion) of bone i. These absolute rotations are just a concatenation down from the root to bone i of the relative rotations we’re storing for each bone. These relative rotations are what we’re changing during an animation.

The four major representations of 3D rotations are rotation matrix, Euler angle (e.g., roll-pitch-yaw), axis-angle (which is very similar to the rotation vector representation), and quaternion. All representations are somewhat equivalent in that they can be converted to a rotation matrix and back again. Feb 05, 2018 · Quaternions were invented in 1843 by William Hamilton, who was so inspired by his discovery that he carved the formula into a bridge nearby. Just as a complex number in 2D contains a real and imaginary component, a quaternion in 4D contains three imaginary components, x , y and z (sometimes a , b and c ), and a real component, w (sometimes d ).

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