- Euler angles give a simple, but perhaps inadequate, parameterization
of orientation
- Euler angles construct a general rotation by a sequence of
rotations about 3 mutually orthogonal axes (x-, y-, z-rolls)
- The order of rolls leads to different parameterizations: we
choose x, y, z order and roll by angles
- Let
denote the cosines and
sines of these three angles respectively - Compositing the well-know principal rotation matrices produces
the parameterization matrix
- There are two drawbacks to the use of Euler angles
that occur because the interaction between rolls about
separate axes
- Gimbal lock is one problem
- A gimbal is a device that permits a body to incline freely
in any direction, or suspend so it remains level when its support
is tipped (Webster's)
- When a degree of freedom is lost, the gimbals is said to ``lock''
- Consider a y-roll of 90 degrees which aligns the x and z axis
- The parameterization matrix applied to a point will negate
the z coordinate, that is, we can not rotate about z
- Interpolation using Euler angles is the second problem
- The problem is that there are multiple rotation paths from one
key position to another
- Euler's theorem states that there always exist a single axis about
which one can rotate from one orientation to another
- A quaternion (discovered by Hamilton in 1843) is a ``number'' of
the form
where
- We'll use the notation
where
- Quaternions satisfy a number of properties
- Multiplication:
- The conjugate of q is
- The magnitude of q is
- What's important for graphics is:
- If
is a ``pure'' quaternion and
is a unit quaternion, define
- Since q is a unit quaternion we can write
- Which yields
- Rotating a vector
by an angular displacement is
achieved by
- ``Lifting'' to quaternion space, i.e. representing
the angular displacement
by the quaternion
- Performing the similarity transform
- The importance of these observations is that using
quaternions, successive rotations can be easily combined
into a single steady rotation
- For example a y-roll by
followed by a z-roll by
simplifies to an x-roll by
- Gimbal lock disappears
Florida Tech Computer Science
William D. Shoaff
Comments to author:wds@cs.fit.edu
All contents copyright ©, William D. Shoaff
Revised: Tue Oct 15 11:13:07 EST 1996