Technical Background - Computer Animation: Algorithms and Techniques - page 60

Also, notice that in performing rotation, qvq 1 , all effects of magnitude are divided out due to the
multiplication by the inverse of the quaternion. Thus, any scalar multiple of a quaternion represents the
same rotation as the corresponding unit quaternion (similar to how the homogeneous representation of
points is scale invariant).
A concise list of quaternion arithmetic and conversions to and from other representations can be
found in Appendix B.3.4 .

2.2.5 Exponential map representation
Exponential maps, similar to quaternions, represent an orientation as an axis of rotation and an asso-
ciated angle of rotation as a single vector [ 3 ]. The direction of the vector is the axis of rotation and the
magnitude is the amount of rotation. In addition, a zero rotation is assigned to the zero vector, making
the representation continuous at the origin. Notice that an exponential map uses three parameters
instead of the quaternion's four. The main advantage is that it has well-formed derivatives. These
are important, for example, when dealing with angular velocity.
This representation does have some drawbacks. Similar to Euler angles, it has singularities. How-
ever, in practice, these can be avoided. Also, it is difficult to concatenate rotations using exponential
maps and is best done by converting to rotation matrices.

2.3 Summary
Linear transformations represented by 4

4 matrices are a fundamental operation in computer graphics
and animation. Understanding their use, how to manipulate them, and how to control round-off error is
an important first step in mastering graphics and animation techniques.
There are several orientation representations to choose from. The most robust representation of ori-
entation is quaternions, but fixed angle, Euler angle, and axis-angle are more intuitive and easier to
implement. Fixed angles and Euler angles suffer from gimbal lock and axis-angle is not easy to com-
posite, but they are useful in some situations. Exponential maps also do not concatenate well but offer
some advantages when working with derivatives of orientation. Appendix B.3.4 contains useful con-
versions between quaternions and other representations.

References
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Addison-Wesley; 1990.
[3] Grassia FS. Practical Parameterization of Rotations Using the Exponential Map. The Journal of Graphics
Tools 1998;3.3.
[4] Mortenson M. Geometric Modeling. New York: John Wiley & Sons; 1997.
[5] Shoemake K. Animating Rotation with Quaternion Curves. In: Barsky BA, editor. Computer Graphics. Pro-
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[6] Strong G. Linear Algebra and Its Applications. New York: Academic Press; 1980.
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