rotation and translation matrices and vectors

Figure 2, below, illustrates the basic problem. There are two matching point sets or, alternatively, a matching point and two vectors in the model which match two vectors in the scene . There are several different transformations which superimpose the model set on the scene set (or vice versa).

These can be combined to form

representing the point to be transformed as in homogeneous coordinates. If is a rotation matrix in 3D orthogonal space, then and the determinant of is 1. Representing and so on this gives 6 constraint equations,

The first intuitive approach to define a rotation matrix might be the ** fixed axis** method , e.g.

- Translate point to the origin
- Rotate by about the original
**x**-axis (positive rotation is from y to z) - Rotate by about the original
**y**-axis (positive rotation is from z to x) - Rotate by about the original
**z**-axis (positive rotation is from x to y)

This leads to a rotation matrix formed by the concatenation of the matrices for the three single angle rotations about the fixed axes,

Note also that the angles can be recovered, , and .

A second method to define a rotation matrix, illustrated in Figure 2, above, is based on a rotation about an
** arbitrary axis**, , by an angle . ,

and is defined

The rotation matrix is

Re-arranging, we can also express and in terms of the rotation matrix elements.

The * trace* of a square matrix is defined as the sum of the diagonal elements, . N is a * normalisation operator*, .

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