Rotations

Matrix

• \(R^T\,R=\mathrm{I}\)
• \(\mathrm{det}(R)=1\)

• (+) Computationaly convenient
• (-) Non-explicit dof, redundancies

Euler Angles

• 3 angles: \((\alpha,\beta,\gamma)\)

• Composition of rotation around basic axes

• Not unique (x-y-z, y-z-x, x-y-x', x-z-x', ...)

• (+) Meaningfull parameters
• (-) Gimbal-lock

Axis angle

• \(({\bf n}, \theta)\)

• (+) Meaningfull parameters
• (-) No direct composition

Quaternion

• \(q=(x,y,z,w)\)
• \(=\left({\bf n}\,\cos\left(\frac{\theta}{2}\right),\sin\left(\frac{\theta}{2}\right)\right)\)

• (+) Composition and interpolation
• (-) Less intuitive components