| import OAI.MeasureTheory.Falconer.Geometry.ChartCoordinates | |
| import Mathlib.Analysis.InnerProductSpace.Projection.Reflection | |
| namespace OAI | |
| /-! | |
| # Choosing a frame adapted to a failed perpendicular test | |
| Two Householder reflections map a fixed orthonormal pair to a prescribed | |
| one. This places the nonstationary tangent direction in the first chart | |
| coordinate without any loss depending on the cap center. | |
| -/ | |
| namespace Falconer | |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] | |
| theorem orthonormal_pair_isometry {a b w t : E} | |
| (ha : ‖a‖ = 1) (hb : ‖b‖ = 1) (hab : inner ℝ a b = 0) | |
| (hw : ‖w‖ = 1) (ht : ‖t‖ = 1) (hwt : inner ℝ w t = 0) : | |
| ∃ e : E ≃ₗᵢ[ℝ] E, e a = w ∧ e b = t := by | |
| let e₁ := (ℝ ∙ (a-w))ᗮ.reflection | |
| have h₁ : e₁ a = w := Submodule.reflection_sub (ha.trans hw.symm) | |
| have horth : inner ℝ w (e₁ b) = 0 := by | |
| rw [← h₁,e₁.inner_map_map] | |
| exact hab | |
| let e₂ := (ℝ ∙ (e₁ b-t))ᗮ.reflection | |
| have h₂ : e₂ (e₁ b) = t := | |
| Submodule.reflection_sub ((e₁.norm_map b).trans (hb.trans ht.symm)) | |
| have hfix : e₂ w = w := by | |
| apply Submodule.reflection_mem_subspace_eq_self | |
| rw [Submodule.mem_orthogonal_singleton_iff_inner_left,inner_sub_right,horth,hwt] | |
| ring | |
| exact ⟨e₁.trans e₂,by simpa only [LinearIsometryEquiv.trans_apply,h₁] using hfix,h₂⟩ | |
| theorem exists_adapted_chart_frame {n : ℕ} {w t : Space (n+2)} | |
| (hw : ‖w‖ = 1) (ht : ‖t‖ = 1) (hwt : inner ℝ w t = 0) : | |
| ∃ e : Space (n+2) ≃ₗᵢ[ℝ] Space (n+2), e (chartPole n) = w ∧ e (chartAxis n) = t := | |
| orthonormal_pair_isometry (chartPole_norm n) (chartAxis_norm n) | |
| (chartPole_inner_axis n) hw ht hwt | |
| theorem inner_isometry_pullback (e : E ≃ₗᵢ[ℝ] E) (x v : E) : | |
| inner ℝ (e x) v = inner ℝ x (e.symm v) := by | |
| rw [← e.inner_map_map x (e.symm v),e.apply_symm_apply] | |
| theorem normalizedLine_isometry (e : E ≃ₗᵢ[ℝ] E) (a t : E) (u : ℝ) : | |
| normalizedLine (e a) (e t) u = e (normalizedLine a t u) := by | |
| simp only [normalizedLine,← e.map_smul,← e.map_add,e.norm_map] | |
| end Falconer | |
| end OAI | |