Definition 4.3.1. Translation.
An transformation \(T\) is a translation if and only if there exists a non-zero constant vector \(\vec{v}\) such that \(T(P)-P=\vec{v}\) for all points \(P.\)
Section 4.3 Algebra of Transformations
Subsection 4.3.1 Explore
Definition 4.3.2. Rotation.
An transformation \(T\) is a rotation if and only if there exists a fixed point \(C\) and constant angle \(\alpha\) such that \(m\angle PCT(P) = \alpha \) and \(\|\overline{CP}\| = \|\overline{CT(P)}\|\) for all points \(P\)
Definition 4.3.3. Reflection.
An transformation \(T\) is a reflection if and only if there exists a fixed line \(\ell\) such that the line perpendicular to \(\ell\) through \(P\) contains \(T(P)\) and the distances from \(P\) and \(T(P)\) to the \(\ell\) are equal.
Theorem 4.3.4.
Translations, rotations, and reflections are isometries.
Checkpoint 4.3.5.
Draw an arbitrary triangle \(ABC.\) Draw the result \(\triangle A'B'C'\) of some translation. Draw the result \(\triangle A''B''C''\) of some translation applied to \(\triangle A'B'C'.\) Determine which type of isometry would transform \(\triangle ABC \) to \(\triangle A''B''C''.\)
Checkpoint 4.3.6.
Complete the following table of composition of isometries.
| Translate | Reflect | Rotate | |
| Translate | |||
| Reflect | |||
| Rotate |
Subsection 4.3.2 Prove
Checkpoint 4.3.7.
How many isometry types are there?
Checkpoint 4.3.8.
How many isometry types are needed to generate all isometry types?
Checkpoint 4.3.9.
How many isometries are needed to generate all isometries?
