Early undergraduateGeometry & TopologyPlanned

Transformation geometry and the Erlangen programme

Classifying the isometries of the plane, affine and projective transformations, and Klein's idea that a geometry is determined by a group of transformations and its invariants.

math.geom.transformation-geometry

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Builds on

Euclidean geometry and proof Linear algebra Algebraic structures: monoids, groups and rings

Required prerequisites, all the way down: 29

You will learn to

Prove that every isometry of the plane is a translation, a rotation, a reflection or a glide reflection.
Represent isometries and affine maps by matrices and compose them.
Explain Klein's Erlangen programme and compare Euclidean, affine and projective invariants.

Definition

An affine map of ℝⁿ has the form x ↦ Ax + b with A an invertible matrix.
A glide reflection is a reflection followed by a translation parallel to the mirror line.
Projective transformations of the real projective plane are induced by invertible 3 × 3 matrices, taken up to non-zero scalar multiples.

Theorem

Every isometry of ℝ² is a translation, a rotation, a reflection or a glide reflection.
Affine maps preserve parallelism and ratios of lengths along a line, but in general not lengths or angles.
Projective transformations preserve the cross-ratio of four collinear points.

Common misconceptions

Assuming every isometry fixes some point; translations and glide reflections do not.
Thinking affine maps preserve angles.

Optional discoveries

Categories Klein's idea can be recast categorically: a group is a one-object category in which every morphism is invertible, and a geometry studies what such morphisms preserve.
Groups Plane patterns repeating in two independent directions fall into exactly 17 symmetry types (the wallpaper groups).
← Rigid motions, congruence and similarity