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The Lumenary
本科低年级Geometry & Topology计划中

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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前置知识

Euclidean geometry and proofLinear algebraAlgebraic structures: monoids, groups and rings

29 项必需的前置知识（一直追溯到底）

你将学会

  • 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.

定义

  • 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.

定理

  • 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.

常见误解

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

可选发现

  • 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

延伸阅读

  • H. S. M. Coxeter, Introduction to Geometry (1961)