Def: Transformation - a bijection between two sets.
Def: Product of transformations (composition of functions), identity, inverses.
Def: Group, Group of transformations.
Example: Symmetries of an equilateral triangle.
Example: Dihedral groups (symmetry groups of regular polygons)
Def: Isometry (a transformation which preserves distances). Motions.
Def: Translations. Rotations. (Rigid motions, displacements).
Def: Glide reflections.
Equations for translations and rotations.
Equations for reflections and glide reflections.
Theorem 2.2: If a transformation is a plane motion, then it has equations of the form:
Theorem 2.3: Converse of the above holds.
Matrix form of equations.
Scale changes, reflections, rotations, shears.
Translations can not be done with 2x2 matrices.
Def: Direct and opposite motions.
Theorem 2.4 : A motion of the plane is uniquely determined by an isometry of one triangle onto another.
Theorem 2.5: The 4 Euclidean motions of the plane constitute a group of transformations.
Theorem 2.6: If a transformation is a plane motion, then it is the product of three or fewer reflections. The converse is also true.
Theorem 2.7: Each finite group of isometries is either a cyclic group or a dihedral group.