Lecture Notes 3

Def: Mapping (function), Domain, Codomain, Range, Onto, One-to-one.

Def: Transformation - a bijection between two sets.

Def: Product of transformations (composition of functions), identity, inverses.

Application: Computer Graphics

World Coordinates to Screen Coordinates

Def: Group, Group of transformations.

Example: Symmetries of an equilateral triangle.

Permutation notation.

Def: Subgroup

Example: Dihedral groups (symmetry groups of regular polygons)

Def: Isometry (a transformation which preserves distances). Motions.

Def: Translations. Rotations. (Rigid motions, displacements).

Def: Reflections.

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:

x' = ax + by + c
y' = (-bx + ay) + d
or
x' = ax + by + c
y' = -(-bx + ay) + d
for a, b, c and d real numbers satisfying a2 + b2 = 1.

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.