Maths - Rotation conversions

Definition of terms:

EuclideanSpace

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Matrix vs. Quaternion

If you need to work with rotations in your program is it best to use maricies or quaternions?

Some advantages of quaternions are:

Each quaternion only requires 4 scalars whereas a matrix requires 9 scalars.
It is easier to interpolate between quaternions using SLERP as explained on this page.
Quaternions are easier to normalise than matrices (to cancel out a build up of small rounding errors).

Some advantages of matrices are:

Transforming a point seems simpler by multiplying a vector by a matrix rather than the sandwich form required for quaternions.
People often find matrices easier to understand than quaternions.
There are more ready built matrix libraries than quaternion libraries.

Isometries and Physics

If we want to model isometries, such as the movement of solid bodies, combining rotation and translation, in one single operation, we need to expand the above algebras to model translations as well as rotations.

To model isometries we could either:

Expand matrices to 4x4 as described on this page.
Expand quaternions to dual quaternions as described on this page.

To investigate these issues further see This page.

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Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

If you are interested in 3D games, this looks like a good book to have on the shelf. If, like me, you want to have know the theory and how it is derived then there is a lot for you here. Including - Graphics pipeline, scenegraph, picking, collision detection, bezier curves, surfaces, key frame animation, level of detail, terrain, quadtrees & octtrees, special effects, numerical methods. Includes CDROM with code.
3D Game engine design also includes conversions between matrix, quaternions and axis-angle.
Mathematics for 3D game Programming - Includes introduction to Vectors, Matrices, Transforms and Trigonometry. (But no euler angles or quaternions). Also includes ray tracing and some linear & rotational physics also collision detection (but not collision response).

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