Summary - Correspondance quaternion / matrix-vector

	3D space	Quaternion space
Vector	\(v=(v_x,v_y,v_z)\)	\(q_v=(v,0)=(v_x,v_y,v_z,0)\)
Rotation	\(R\) (\(3\times 3\) matrix)	\(q=(x,y,z,w)\), \(\|q\|=1\)
Apply rotation to vector	\(R v\)	\(q\;q_v\;q^{\star}\)
Rotation composition	\(R_1\; R_2\)	\(q_1\;q_2\)

Rotation to Quaternion

• Rotation of axis \(n\) and angle \(\theta\) \(\Rightarrow q=(n\sin(\theta/2),\cos(\theta/2))\)

Quaternion to Rotation

• Unit quaternion \(q=(x,y,z,w)\) \(\Rightarrow R= \left( \begin{array}{ccc} 1-2(y ^ 2+z ^ 2) & 2 (x y - w z) & 2 (x z + w y) \\ 2 (x y + w z) & 1 - 2 (x ^ 2 + z ^ 2) & 2 (y z - w x) \\ 2 (x z - w y) & 2 (y z + w x) & 1- 2 (x ^ 2 + y ^ 2) \end{array} \right)\)