Summary

1. Equivalent in power to Gibbs vectors, with dot and cross products.
2. Easier than vectors for 3D rotations, e.g.,
   1. combining multiple rotations
   2. animating a rotation
3. other possibilities for rotations:
   1. matrices
   2. Euler_Angles - very bad for animation, and suffer from Gimbal_lock
   3. Rotation3D
4. Invented by WR Hamilton in 1843 as a 3D analog to complex numbers, to do 3D rotations. However, what we use today is a distilled essence of the complicated way that Hamilton presented them. (That is true for many mathematical ideas.)

Refs

1. my paper
2. Angel, page 224ff.
3. http://en.wikipedia.org/wiki/Quaternion
4. http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation
5. http://en.wikipedia.org/wiki/History_of_quaternions
6. The Matrix and Quaternions FAQ
7. Rotations in Three Dimensions, Part Five: Quaternions
8. many other places

Sample question

What rotation does this quaternion represent: q = 0.707 + 0.707 j

Answer: a 90^\circ rotation about the axis (0,1,0)

Examples

Start with some 2D geometry and complex number examples.

Let the point p=(1,2). The corresponding complex number is c=1+2i. Suppose you want to rotate it by \theta=90^\circ=\pi/2 radians. That is equivalent to multiplying c by e^{i\theta}=e^{i\pi/2}=i.

So, c'=c e^{i\theta} = (1+2i)i = -2+i.

The corresponding 2D point is (-2,1).

Now to quaternions in general.

Let q_1=(1,2,0,0) and q_2=(3,0,4,0).
q_1+q_2=(4,0,4,0).

Now to 3D and quaternions.

For example, the 3D point (1,0,2) corresponds to the quaternion 1i+0j+2k.

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