Analytic Foundations of Geometry

Robert S. Wilson

4. Reflections and Translations

Definitions

An isometry is a function that maps the plane to itself which preserves distances. If M is an isometry, we will denote the point to which M moves a point A by M(A). When we say that an isometry preserves distances we mean that

|M(A)M(B)| = |AB|

An isometry M is said to be invertible if there is another isometry M-1 such that

M(M-1(A)) = M-1(M(A)) = A

for every point A in the plane.

A translation is a map of the form

T((x , y) = (x - x0 , y - y0)

for some point (x0, y0) in the plane.

Given a line, to reflect a point across the line, move the point perpendicularly to the line and then move it the same distance beyond the line. Such a map is called a reflection.