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All isometries of space can be expressed in homogeneous coordinates in
terms of multiplication by a matrix. As in the case of plane
isometries (Section 2.2), this means that the successive
application of transformations reduces to matrix multiplication. (In
the formulas below, is the 4×4
projective matrix obtained from the 3×3 matrix *M* by
adding a row and a column as stated.)

**Translation** by (*x*,*y*,*z*):

**Rotation** through the origin:
where *M* is given in (10.1.1)
or (10.1.2)
, as
the case may be.

**Reflection** in a plane through the origin:
where *M* is given in (10.1.3)
.

From this one can deduce all other transformations, as in the case of plane transformations .

*Silvio Levy
Wed Oct 4 16:41:25 PDT 1995*

This document is excerpted from the 30th Edition of the *CRC Standard Mathematical Tables and Formulas* (CRC Press). Unauthorized duplication is forbidden.