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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):
Rotation through (counterclockwise) around the origin:
Rotation through (counterclockwise) around an arbitrary point (x,y):
Reflection in the x-axis:
Reflection in the y-axis:
Reflection in the xy-diagonal:
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.
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.