####

Page 6 -- 3D Transformations Using A 3x3 Matrix

Transformations in three dimensions are generally used to describe pure translation, pure rotation, rotation about any axis, or a combination of these three processes for a vector or set of vectors.

**PURE TRANSLATION**

*To change the position of the origin of a coordinate system but maintain attitude in three dimensions*

TRANSLATION DEMO

**PURE ROTATION**

*To rotate the coordinate system about one of the unit axes*
Pure rotation about a coordinate axis in three dimensions is similar to 2D rotation, except that the additional degree of freedom (or the third vector coordinate) must be preserved. A unit marker is placed in the matrix position corresponding to axis rotation, as shown below for each coordinate axis.

PURE ROTATION DEMO

**ROTATION ABOUT ANY AXIS**

*To rotate the coordinate system about a vector {u,v,w}*
Of course it is possible to rotate the coordinate system about any axis in space, with a rotation matrix of the following form:

where "c" and "s" indicate cosine and sine functions, respectively, and {u,v,w} is the vector notation of the rotation axis in global coordinates.

AXIS ROTATION DEMO