Geometric Transformations

Definitions

Definitions

Examples

Examples

Geometric Transformations

Geometric Transformations

Composition of Transformations

Example

An Application: Computer Graphics

Groups

Examples

Symmetries of an Equilateral Triangle

Symmetries of an Equilateral Triangle

Symmetries of Triangle form a Group

Symmetries of Triangle form a Group

Symmetries of Triangle form a Group

Subgroups

Symmetries of Triangle form a Group

Dihedral Groups

Isometry

Translations

Translations

Translations

Rotations

Rotations

Translations and Rotations

Reflections

Reflections

Glide Reflections

No Quiz Thursday

Equations for Rigid Motions

Equations for Rigid Motions

Equations for Rigid Motions

Rotations are Isometries

Equations for Reflections

Equations for Reflections

Equations for Reflections

Equations for Reflections

Equations for Reflections

Theorem 2.2

Theorem 2.3

Matrix Form of Equations

Matrix Form of Equations

More Transformations

Translations

Direct and Opposite Motions

Direct and Opposite Motions

Direct and Opposite Motions

Some Theorems

Proof of Theorem 2.6

Proof of Theorem 2.6

Proof of Theorem 2.6

Proof of Theorem 2.6

Proof of Theorem 2.6

Proof of Theorem 2.6

Proof of Theorem 2.6

Proof of Theorem 2.6

Proof of Theorem 2.6

Proof of Theorem 2.6

Theorem 2.6

Cyclic Groups

Theorem 2.7