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

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