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Mathlib.Geometry.Euclidean.Inversion (#5868)
Move theorems about image of a sphere passing through the center to a new file.
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| Original file line number | Diff line number | Diff line change |
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| /- | ||
| Copyright (c) 2023 Yury Kudryashov. All rights reserved. | ||
| Released under Apache 2.0 license as described in the file LICENSE. | ||
| Authors: Yury Kudryashov | ||
| -/ | ||
| import Mathlib.Geometry.Euclidean.Inversion.Basic | ||
| import Mathlib.Geometry.Euclidean.PerpBisector | ||
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| /-! | ||
| # Image of a hyperplane under inversion | ||
| In this file we prove that the inversion with center `c` and radius `R ≠ 0` maps a sphere passing | ||
| through the center to a hyperplane, and vice versa. More precisely, it maps a sphere with center | ||
| `y ≠ c` and radius `dist y c` to the hyperplane | ||
| `AffineSubspace.perpBisector c (EuclideanGeometry.inversion c R y)`. | ||
| The exact statements are a little more complicated because `EuclideanGeometry.inversion c R` sends | ||
| the center to itself, not to a point at infinity. | ||
| We also prove that the inversion sends an affine subspace passing through the center to itself. | ||
| ## Keywords | ||
| inversion | ||
| -/ | ||
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| open Metric Function AffineMap Set AffineSubspace | ||
| open scoped Topology | ||
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| variable {V P : Type _} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] | ||
| [NormedAddTorsor V P] {c x y : P} {R : ℝ} | ||
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| namespace EuclideanGeometry | ||
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| /-- The inversion with center `c` and radius `R` maps a sphere passing through the center to a | ||
| hyperplane. -/ | ||
| theorem inversion_mem_perpBisector_inversion_iff (hR : R ≠ 0) (hx : x ≠ c) (hy : y ≠ c) : | ||
| inversion c R x ∈ perpBisector c (inversion c R y) ↔ dist x y = dist y c := by | ||
| rw [mem_perpBisector_iff_dist_eq, dist_inversion_inversion hx hy, dist_inversion_center] | ||
| have hx' := dist_ne_zero.2 hx | ||
| have hy' := dist_ne_zero.2 hy | ||
| field_simp [mul_assoc, mul_comm, hx, hx.symm, eq_comm] | ||
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| /-- The inversion with center `c` and radius `R` maps a sphere passing through the center to a | ||
| hyperplane. -/ | ||
| theorem inversion_mem_perpBisector_inversion_iff' (hR : R ≠ 0) (hy : y ≠ c) : | ||
| inversion c R x ∈ perpBisector c (inversion c R y) ↔ dist x y = dist y c ∧ x ≠ c := by | ||
| rcases eq_or_ne x c with rfl | hx | ||
| · simp [*] | ||
| · simp [inversion_mem_perpBisector_inversion_iff hR hx hy, hx] | ||
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| theorem preimage_inversion_perpBisector_inversion (hR : R ≠ 0) (hy : y ≠ c) : | ||
| inversion c R ⁻¹' perpBisector c (inversion c R y) = sphere y (dist y c) \ {c} := | ||
| Set.ext fun _ ↦ inversion_mem_perpBisector_inversion_iff' hR hy | ||
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| theorem preimage_inversion_perpBisector (hR : R ≠ 0) (hy : y ≠ c) : | ||
| inversion c R ⁻¹' perpBisector c y = sphere (inversion c R y) (R ^ 2 / dist y c) \ {c} := by | ||
| rw [← dist_inversion_center, ← preimage_inversion_perpBisector_inversion hR, | ||
| inversion_inversion] <;> simp [*] | ||
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| theorem image_inversion_perpBisector (hR : R ≠ 0) (hy : y ≠ c) : | ||
| inversion c R '' perpBisector c y = sphere (inversion c R y) (R ^ 2 / dist y c) \ {c} := by | ||
| rw [image_eq_preimage_of_inverse (inversion_involutive _ hR) (inversion_involutive _ hR), | ||
| preimage_inversion_perpBisector hR hy] | ||
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| theorem preimage_inversion_sphere_dist_center (hR : R ≠ 0) (hy : y ≠ c) : | ||
| inversion c R ⁻¹' sphere y (dist y c) = | ||
| insert c (perpBisector c (inversion c R y) : Set P) := by | ||
| ext x | ||
| rcases eq_or_ne x c with rfl | hx; · simp [dist_comm] | ||
| rw [mem_preimage, mem_sphere, ← inversion_mem_perpBisector_inversion_iff hR] <;> simp [*] | ||
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| theorem image_inversion_sphere_dist_center (hR : R ≠ 0) (hy : y ≠ c) : | ||
| inversion c R '' sphere y (dist y c) = insert c (perpBisector c (inversion c R y) : Set P) := by | ||
| rw [image_eq_preimage_of_inverse (inversion_involutive _ hR) (inversion_involutive _ hR), | ||
| preimage_inversion_sphere_dist_center hR hy] | ||
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| /-- Inversion sends an affine subspace passing through the center to itself. -/ | ||
| theorem mapsTo_inversion_affineSubspace_of_mem {p : AffineSubspace ℝ P} (hp : c ∈ p) : | ||
| MapsTo (inversion c R) p p := fun _ ↦ AffineMap.lineMap_mem _ hp | ||
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| /-- Inversion sends an affine subspace passing through the center to itself. -/ | ||
| theorem image_inversion_affineSubspace_of_mem {p : AffineSubspace ℝ P} (hR : R ≠ 0) (hp : c ∈ p) : | ||
| inversion c R '' p = p := | ||
| (mapsTo_inversion_affineSubspace_of_mem hp).image_subset.antisymm <| fun x hx ↦ | ||
| ⟨inversion c R x, mapsTo_inversion_affineSubspace_of_mem hp hx, inversion_inversion _ hR _⟩ |