** Next:** 2.2 Formulas for Symmetries in Homogeneous Coordinates
**Up:** 2 Plane Symmetries or Isometries
** Previous:** 2 Plane Symmetries or Isometries

In the formulas below, a multiplication between a matrix and a pair of
coordinates should be carried out regarding the pair as a column
vector (or a matrix with two rows and one column). Thus
(*x*,*y*)=(*ax*+*by*, *cx*+*dy*).

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

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

**Rotation** through (counterclockwise) around the origin:

**Rotation** through (counterclockwise)
around an arbitrary point (*x*,*y*):

**Reflection** in the *x*-axis:

(*x*,*y*)(*x*,-*y*)

**Reflection** in the *y*-axis:

(*x*,*y*)(-*x*,*y*)

**Reflection** in the *xy*-diagonal:

(*x*,*y*)(*y*,*x*)

**Reflection** in a line with equation *ax*+*by*+*c*=0:

**Reflection** in a line going through (*x*,*y*) and making an
angle with the *x*-axis:

**Glide-reflection** in a line *L* with displacement *d*: Apply first
a reflection in *L*, then a translation by a vector of length *d* in
the direction of *L*, that is, by the vector

if *L* has equation *ax*+*by*+*c*=0.

*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.