Fixed-Axis Rotations, Quaternions

PREVIOUS < - - - - > CS 184 HOME < - - - - > CURRENT < - - - - > NEXT

How to describe or specify arbitrary rotations ?

Specifying rotations with Euler angles can lead to problems in animation and in robotics, because the coordinate axes are singled out in a specific way. For instance, flying a plane over the North Pole may suddenly flip it around; or moving some mechanical linkage through multiple rotational operations may get it stuck in an awkward configuration, experiencing 'gimbal lock.'

To avoid these problems we need a representation for angles and rotations that is less dependent on any coordinate system.
One such system is based on the "axis+angle" representation and can be realized with 'quaternions.'
Quaternions are normalized 4-vectors, where the four components encode the angle of rotation and the three components of the rotation axis.
Specifically: the quaternion representing a rotation through an angle a around an axis of rotation with components rx, ry, rz of a normalized vector in the axis direction is given by
Q = [ cos(a/2), sin(a/2)*rx, sin(a/2)*ry, sin(a/2)*rz ].

Its inverse is given by:
Q^-1 = [ cos(a/2), -sin(a/2)*rx, -sin(a/2)*ry, -sin(a/2)*rz ].
This can be understood as a rotation in the opposite direction. Rather than negating the rotation angle, the rotation axis has been reversed.

To rotate some geometry by this quaternion, each vertex is first turned into a quaternion : P ==> Q_p = [ 0, px, py, pz ]
and it is then rotated by forming the quaternion product:
Q'_p = Q * Q_p * Q^-1

See: Shirley 2nd Edition, Ch. 16.2.2 or Shirley 3rd Edition, Ch. 17.2.2

PREVIOUS < - - - - > CS 184 HOME < - - - - > CURRENT < - - - - > NEXT