Shahada ya kwanza (mwanzo)Geometry & TopologyImepangwa
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
Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.
Inajengwa juu ya
Masharti ya awali ya lazima 29 (hadi chini kabisa)
Utajifunza
- 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.
Fasili
- 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.
Nadharia
- 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.
Mawazo potofu ya kawaida
- Assuming every isometry fixes some point; translations and glide reflections do not.
- Thinking affine maps preserve angles.
Ugunduzi wa hiari
- 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
Usomaji zaidi
- H. S. M. Coxeter, Introduction to Geometry (1961)