Practical Parameterization of Rotations using Exponential Maps

Summary:

This paper discusses several ways to represent rotation that can be used
in computer animation and their merits and faults. A particular focus is
given to exponential maps and an explanation of why they are near ideal
for rotation calculations in use in modern computer animation.

Problem:

This paper is exploring the problem of how to best represent rotations in
three dimensional space for the purposes of computer animation. The author
specifically lists out the requirements that modern computer animation has
for a rotation system before beginning to explore this problem.

Method:

This paper considers the weaknesses of several forms of rotation
representation by explaining how they measure up to the six basic
primitives that are required for describing and controlling rotations in
computer graphics. Special focus is given to exponential maps and their
advantages when computing each of the primitives.

Key Ideas:

The exponential maps discussed in this paper are essentially a mapping of
quaternions (4D numbers) to an axis of rotation in 3D and the rotation's
magnitude. Since the exponential map can only represent a rotations less
than 360 degrees unambiguously. Fortunately if you measure relative
rotation between rendering intervals, the change is almost always less
than a full rotation (since human eyes can't disambiguate this much
movement either!) so the singularity problem can be avoided. Also the
exponential maps have other desirable features such as simple derivative

calculations and a cheap/simple conversion to and from quaternions.
Because some primitives can be calculated without converting the
exponential map back to a quaternion, it will be computationally cheaper
on average to use the exponential map rather than raw quaternions while
still allowing conversion for primitives which quaternions excel at (such
as interpolation).
Other mathematically simple representations, such as Euler angles suffer
from far more restrictive singularities and can not interpolate in a
smooth, accurate manner.

Contributions:

Exponential maps are a significant contribution because they are more
efficient than quaternions but are still able to take advantage of the
interpolative powers of quaternions.

Questions:

I am curious about how often exponential maps are used in realtime
applications verses less dynamic applications, such as modeling programs.