Math 2.0

Geometry Theorem

Classification of isometries of the plane

Builds on Euclidean isometryTranslationRotationReflectionGlide reflection

Statement

Let ff be a Euclidean isometry of the plane ℝ2\mathbb{R}^2. Then ff is of exactly one of these five types:

Remarks

Write f(x)=Ax+bf(x) = A x + b with AA orthogonal. If det⁡A=1\det A = 1, then ff is the identity, a translation or a rotation. If det⁡A=−1\det A = -1, then ff is a reflection or a glide reflection. The fixed points tell the types apart. A translation and a glide reflection have no fixed point. A rotation has one, and a reflection fixes a whole line. Each isometry of the plane is the composite of at most three reflections in lines.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Euclidean isometry, concept, no evidence levelEuclidean isometryconceptGlide reflection, concept, no evidence levelGlide reflectionconceptReflection, concept, no evidence levelReflectionconceptRotation, concept, no evidence levelRotationconceptTranslation, concept, no evidence levelTranslationconceptClassification of isometries of the plane, statement, no evidence levelClassification of iso…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean isometry concept
  2. Glide reflection concept
  3. Reflection concept
  4. Rotation concept
  5. Translation concept
  6. Classification of isometries of the plane statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Euclidean_plane_isometry, Classification
  2. Web https://en.wikipedia.org/wiki/Euclidean_plane_isometry, Recognition
  3. Web https://en.wikipedia.org/wiki/Euclidean_plane_isometry, Isometries as reflection group
About this page
Tag0P0XDY9FZN
URIm2://0P0XDY9FZN
Kindstatement, theorem
WritingW0 stub
MSC51F20
StatusThis copy of the site has no build data. Run m2 status 0P0XDY9FZN.
UsesEuclidean isometry, Translation, Rotation, Reflection, Glide reflection

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