Primeros años de universidadGeometry & TopologyPlanificado
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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Se basa en
Requisitos previos obligatorios, hasta el fondo: 29
Aprenderás a
- 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.
Definición
- 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.
Teorema
- 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.
Errores frecuentes
- Assuming every isometry fixes some point; translations and glide reflections do not.
- Thinking affine maps preserve angles.
Descubrimientos opcionales
- 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
Lecturas complementarias
- H. S. M. Coxeter, Introduction to Geometry (1961)