# mathlibdocumentation

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 #

noncomputable def rotation  :

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

Equations
@[simp]
theorem rotation_apply (a : circle) (z : ) :
@[simp]
theorem rotation_symm (a : circle) :
@[simp]
theorem rotation_trans (a b : circle) :
noncomputable def rotation_of (e : ≃ₗᵢ[] ) :

Takes an element of ℂ ≃ₗᵢ[ℝ] ℂ and checks if it is a rotation, returns an element of the unit circle.

Equations
@[simp]
theorem rotation_of_rotation (a : circle) :
= a
theorem linear_isometry.re_apply_eq_re_of_add_conj_eq (f : →ₗᵢ[] ) (h₃ : ∀ (z : ), z + z = f z + (f z)) (z : ) :
(f z).re = z.re
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.im_apply_eq_im {f : →ₗᵢ[] } (h : f 1 = 1) (z : ) :
z + z = f z + (f z)
theorem linear_isometry.re_apply_eq_re {f : →ₗᵢ[] } (h : f 1 = 1) (z : ) :
(f z).re = z.re
theorem to_matrix_rotation (a : circle) :

The matrix representation of rotation a is equal to the conformal matrix !![re a, -im a; im a, re a].

@[simp]
theorem det_rotation (a : circle) :

The determinant of rotation (as a linear map) is equal to 1.

@[simp]

The determinant of rotation (as a linear equiv) is equal to 1.