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In the formulas below, a multiplication between a matrix and a triple of coordinates should be carried out regarding the triple as a column vector (or a matrix with three rows and one column).

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

(*x*,*y*,*z*)(*x*+*x*, *y*+*y*, *z*+*z*)

**Rotation** through (counterclockwise) around the line
through the origin with direction cosines *a*, *b*, *c*
(see ):

(*x*,*y*,*z*)*M*(*x*,*y*,*z*),

where *M* is the matrix

**Rotation** through (counterclockwise) around the line
with direction cosines *a*,*b*,*c* through an arbitrary point (*x*,*y*,*z*):

(*x*,*y*,*z*)(*x*,*y*,*z*)+*M*(*x*-*x*,
*y*-*y*, *z*-*z*),

where *M* is given by (1)
.

**Arbitrary rotations and Euler angles.** Any rotation of space
fixing the origin can be decomposed as a rotation by about the
*z*-axis, followed by a rotation by about the *y*-axis,
followed by rotation by about the *z*-axis. The numbers
, and are called the **Euler angles** of the
composite rotation, which acts as follows:

(*x*,*y*,*z*)*M*(*x*,*y*,*z*),

where *M* is the matrix given by

(An alternative decomposition, more natural if we think of the
coordinate system as a rigid trihedron that rotates in space, is the
following: a rotation by about the *z*-axis, followed by a
rotation by about the *rotated* *y*-axis, followed by a
rotation by about the *rotated* *z*-axis. Note that the
order is reversed.)

Provided that is not a multiple of 180°, the decomposition of a rotation in this form is unique (apart from the ambiguity arising from the possibility of adding a multiple of 360°to any angle). Figure 1 shows how the Euler angles can be read off geometrically.

**Figure 1:** The coordinate rays *Ox*, *Oy*, *Oz*, together with their images
*O*, *O*, *O* under a rotation, fix the Euler angles associated
with that rotation, as follows:
=*zO*, =*xOr*=*yOs*, and =*sO*. (Here
the ray *Or* is the projection of *O* to the
*xy*-plane. The ray *Os* is determined by the intersection of the
*xy*- and -planes.)

**Warning.** Some references define Euler angles differently; the most
common variation is that the second rotation is taken about the
*x*-axis instead of about the *y*-axis.

**Screw motion** with angle and displacement *d* around the line
with direction cosines *a*, *b*, *c* through an arbitrary point (*x*,*y*,*z*):

(*x*,*y*,*z*)(*x*+*ad*, *y*+*bd*, *z*+*cd*) + *M*(*x*-*x*, *y*-*y*, *z*-*z*),

where *M* is given by (1)
.

**Reflection**

**Reflection** in a plane with equation *ax*+*by*+*cz*+*d*=0:

**Reflection** in a plane going through (*x*,*y*,*z*) and whose
normal has direction cosines *a*, *b*, *c*:

(*x*,*y*)(*x*+*y*+*z*)+*M*(*x*-*x*, *y*-*y*,
*z*-*z*),

where *M* is as in (3)
.

**Glide-reflection** in a plane *P* with displacement vector
**v**: Apply first a reflection in *P*, then a translation by
the vector **v**.

*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.