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:

1. show that the real parts match up: LinearIsometry.re_apply_eq_re
2. show that I maps to either I or -I
3. every z is a linear combination of a + b * I

References

• Isometries of the Complex Plane

noncomputable def rotation : Circle →* ℂ ≃ₗᵢ[ℝ] ℂ

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

@[simp]

theorem rotation_apply (a : Circle) (z : ℂ) : (rotation a) z = ↑a * z

@[simp]

theorem rotation_symm (a : Circle) : (rotation a).symm = rotation a⁻¹

@[simp]

theorem rotation_trans (a b : Circle) : (rotation a).trans (rotation b) = rotation (b * a)

theorem rotation_ne_conjLIE (a : Circle) : rotation a ≠ Complex.conjLIE

noncomputable def rotationOf (e : ℂ ≃ₗᵢ[ℝ] ℂ) : Circle

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

@[simp]

theorem rotationOf_coe (e : ℂ ≃ₗᵢ[ℝ] ℂ) : ↑(rotationOf e) = e 1 / ↑‖e 1‖

@[simp]

theorem rotationOf_rotation (a : Circle) : rotationOf (rotation a) = a

theorem rotation_injective : Function.Injective ⇑rotation

theorem LinearIsometry.re_apply_eq_re_of_add_conj_eq (f : ℂ →ₗᵢ[ℝ] ℂ) (h₃ : ∀ (z : ℂ), z + (starRingEnd ℂ) z = f z + (starRingEnd ℂ) (f z)) (z : ℂ) : (f z).re = z.re

theorem LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re {f : ℂ →ₗᵢ[ℝ] ℂ} (h₂ : ∀ (z : ℂ), (f z).re = z.re) (z : ℂ) : (f z).im = z.im ∨ (f z).im = -z.im

theorem LinearIsometry.im_apply_eq_im {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ) : z + (starRingEnd ℂ) z = f z + (starRingEnd ℂ) (f z)

theorem LinearIsometry.re_apply_eq_re {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ) : (f z).re = z.re

theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) : f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = Complex.conjLIE

theorem linear_isometry_complex (f : ℂ ≃ₗᵢ[ℝ] ℂ) : ∃ (a : Circle), f = rotation a ∨ f = Complex.conjLIE.trans (rotation a)

theorem toMatrix_rotation (a : Circle) : (LinearMap.toMatrix Complex.basisOneI Complex.basisOneI) ↑(rotation a).toLinearEquiv = ↑(Matrix.planeConformalMatrix (↑a).re (↑a).im ⋯)

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) : LinearMap.det ↑(rotation a).toLinearEquiv = 1

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

@[simp]

theorem linearEquiv_det_rotation (a : Circle) : LinearEquiv.det (rotation a).toLinearEquiv = 1

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