| import OAI.Geometry.RuledCovering.Projection |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Filter Metric |
| open scoped Topology ContDiff Pointwise |
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| namespace RuledApproximation |
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| abbrev Coordinates := Fin 2 → ℝ |
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| def frameJacobian (M : Coordinates ≃L[ℝ] Plane) : ℝ := |
| |(M.trans (EuclideanSpace.equiv (Fin 2) ℝ)).toContinuousLinearMap.det| |
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| lemma frameJacobian_pos (M : Coordinates ≃L[ℝ] Plane) : 0 < frameJacobian M := by |
| apply abs_pos.mpr |
| exact (M.trans (EuclideanSpace.equiv (Fin 2) ℝ)).toLinearEquiv.isUnit_det'.ne_zero |
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| lemma frame_integral_image (M : Coordinates ≃L[ℝ] Plane) {K : Set Coordinates} |
| (hK : MeasurableSet K) (F : Plane → ℝ) : |
| (∫ p in M '' K, F p) = frameJacobian M * (∫ x in K, F (M x)) := by |
| let e := EuclideanSpace.equiv (Fin 2) ℝ |
| let D := M.trans e |
| have himage : M '' K = e.symm '' (D '' K) := by |
| rw [← image_comp] |
| congr 1 |
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| rw [himage] |
| have hmp : MeasurePreserving e.symm volume volume := PiLp.volume_preserving_toLp (Fin 2) |
| rw [hmp.setIntegral_image_emb e.symm.toHomeomorph.measurableEmbedding] |
| rw [integral_image_eq_integral_abs_det_fderiv_smul volume hK |
| (fun x _ => D.hasFDerivAt.hasFDerivWithinAt) D.injective.injOn] |
| rw [integral_smul] |
| rfl |
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| lemma frame_volume_image (M : Coordinates ≃L[ℝ] Plane) {K : Set Coordinates} |
| (hK : MeasurableSet K) : volume.real (M '' K) = frameJacobian M * volume.real K := by |
| simpa using frame_integral_image M hK (fun _ => (1 : ℝ)) |
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| lemma coordinates_closedBall (b : Coordinates) {a : ℝ} (ha : 0 ≤ a) : |
| closedBall b a = Set.pi univ (fun i => Icc (b i - a) (b i + a)) := by |
| ext x |
| simp only [mem_closedBall, dist_eq_norm, pi_norm_le_iff_of_nonneg ha, Real.norm_eq_abs, |
| Pi.sub_apply, Set.mem_pi, mem_univ, true_implies, mem_Icc, abs_le] |
| constructor <;> intro h i <;> obtain ⟨h₁, h₂⟩ := h i <;> constructor <;> linarith |
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| lemma coordinates_closedBall_volume (b : Coordinates) {a : ℝ} (ha : 0 ≤ a) : |
| volume.real (closedBall b a) = (2 * a) ^ 2 := by |
| rw [coordinates_closedBall b ha, measureReal_def, volume_pi_pi] |
| simp only [Real.volume_Icc, add_sub_sub_cancel, Fin.prod_univ_two] |
| rw [ENNReal.toReal_mul] |
| rw [ENNReal.toReal_ofReal (by positivity : 0 ≤ a + a)] |
| ring |
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| lemma coordinates_closedBall_parallelogram (b : Coordinates) {a : ℝ} (ha : 0 < a) : |
| IsParallelogram (closedBall b a) := by |
| let u : Coordinates := ![2*a, 0] |
| let v : Coordinates := ![0, 2*a] |
| refine ⟨![b 0 - a, b 1 - a], u, v, ?_, ?_⟩ |
| · have heq : (![u, v] : Fin 2 → Coordinates) = |
| fun i => (2 * a) • Pi.basisFun ℝ (Fin 2) i := by |
| funext i |
| ext j |
| fin_cases i <;> fin_cases j <;> simp [u, v, Pi.basisFun_apply] |
| rw [heq] |
| exact (Pi.basisFun ℝ (Fin 2)).linearIndependent.units_smul |
| (fun _ => Units.mk0 (2 * a) (by positivity)) |
| · rw [coordinates_closedBall b ha.le] |
| ext x |
| simp only [Set.mem_pi, mem_univ, true_implies, mem_Icc, Set.mem_ofPred_eq] |
| constructor |
| · intro hx |
| refine ⟨(x 0 - (b 0 - a)) / (2*a), ?_, (x 1 - (b 1 - a)) / (2*a), ?_, ?_⟩ |
| · constructor |
| · exact div_nonneg (sub_nonneg.mpr (hx 0).1) (by positivity) |
| · apply (div_le_one (by positivity)).mpr |
| linarith [(hx 0).2] |
| · constructor |
| · exact div_nonneg (sub_nonneg.mpr (hx 1).1) (by positivity) |
| · apply (div_le_one (by positivity)).mpr |
| linarith [(hx 1).2] |
| · ext i |
| fin_cases i <;> simp [u, v, (by positivity : 2*a ≠ 0)] |
| · rintro ⟨s, hs, t, ht, rfl⟩ i |
| fin_cases i <;> simp [u, v] <;> constructor <;> nlinarith [hs.1, hs.2, ht.1, ht.2] |
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| end RuledApproximation |
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| end |
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| end OAI |
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