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analysis.complex.isometry

Isometries of the Complex Plane #

The lemma linear_isometry_complex states the classification of isometries in the complex plane. Specifically, isometries with rotations but without translation. The proof involves:

  1. creating a linear isometry g with two fixed points, g(0) = 0, g(1) = 1
  2. applying linear_isometry_complex_aux to g The proof of linear_isometry_complex_aux is separated in the following parts:
  3. show that the real parts match up: linear_isometry.re_apply_eq_re
  4. show that I maps to either I or -I
  5. every z is a linear combination of a + b * I

References #

An element of the unit circle defines a linear_isometry_equiv from to itself, by rotation. This is an auxiliary construction; use rotation, which has more structure, by preference.

Equations

An element of the unit circle defines a linear_isometry_equiv from to itself, by rotation.

Equations
@[simp]
theorem rotation_apply (a : circle) (z : ) :
(rotation a) z = (a) * z
theorem linear_isometry.im_apply_eq_im_or_neg_of_re_apply_eq_re {f : →ₗᵢ[] } (h₂ : ∀ (z : ), (f z).re = z.re) (z : ) :
(f z).im = z.im (f z).im = -z.im
theorem linear_isometry.re_apply_eq_re {f : →ₗᵢ[] } (h : f 1 = 1) (z : ) :
(f z).re = z.re