6.1 SYMMETRY OF PLANE FIGURES
Bilateral, rotational, translational, glide symmetry, and their combinations.
6.2 ISOMETRIES
6.2.1 Def. (Distance, Isometry)
The distance between points of \(\mathbb{R}^n\) is the length \(|u-v|\) of the vector \(u-v\). An isometry of \(n\)-dimensional space \(\mathbb{R}^n\) is a distance-preserving map \(f\) from \(\mathbb{R}^n\) to itself, a map such that, \(\forall u \in \mathbb{R}^n\),
\[|f(u) - f(v)|=|u-v|.
\]
An isometry will map a figure to a congruent figure.
6.2.2. Examples.
- (a) Orthogonal liear operators are isometries.
- Orthogonal operator preserve dot product \(\rightarrow\) distance.
- (b) Translation \(t_a(x) = x + a\) by vector \(a\) is isometry.
- (c) The composition of isometries is an isometry.