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d4317d7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 | 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
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