Math 403 Part III. Isometries in the Euclidean Plane.

These lessons are based on Chapter 4 of Philippe Tondeur’s textbook. They differ from Tondeur’s exposition. The techniques and methods of proof are collected into three "super lemmas", whose nicknames are the Barytheorem, Recalibration theorem, and the Conjugacy theorem. And we do not call linear isometries by that name until we have proved that these point transformations of the plane are linear. Until then, and occasionally thereafter, they are called osometries because their defining property is that they preserve the origin of the Cartesian coordinate system imposed on the plane.

• I1 Definition of isometry.

• I2 The Osometry Factorization Theorem

• I3 The Barycentric Coordinates Theorem

• I4 Reflections and their properties.

• I5 Part I Rotations and their properties.

• I5 Part II Translations and their properties.

• I6 Classifications by minimal mirrors.

• I7 Glide-reflections and the Classification Theorem

• I10 Exercises on Isometries