| import OAI.Geometry.CylinderCovering.Simplex.MetricNormalizationDefinitions |
| import OAI.Geometry.CylinderCovering.Simplex.Projection |
| import OAI.Geometry.CylinderCovering.Convexity.CylinderSimilarity |
| import OAI.Geometry.CylinderCovering.Convexity.RegularSimplexSimilarity |
| import OAI.Geometry.TriangularCovering.Main |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory |
|
|
| namespace CylinderCovering |
|
|
| theorem regular_minimum_shadow_area : |
| ∀ v : Fin 4 → Space, AffineIndependent ℝ v → |
| ∀ edge : ℝ, 0 < edge → |
| (∀ i j : Fin 4, i ≠ j → dist (v i) (v j) = edge) → |
| minimumShadowArea (convexHull ℝ (range v)) = edge ^ 2 / (2 * Real.sqrt 2) := by |
| intro v hv edge hedge hdist |
| obtain ⟨L, t, hsim⟩ := RegularSimplexSimilarity.regular_simplex_similarity |
| TriangularCovering.tetraVertices v TriangularCovering.tetra_affineIndependent hv |
| (show (0 : ℝ) < 2 by norm_num) hedge |
| (fun _ _ h => TriangularCovering.tetra_edge_length h) hdist |
| rw [← TriangularCovering.K_eq_convexHull] at hsim |
| rw [hsim] |
| have hr : edge / 2 ≠ 0 := ne_of_gt (div_pos hedge (by norm_num : (0 : ℝ) < 2)) |
| change Similarity.minimumArea (Similarity.map t (edge / 2) L '' TriangularCovering.K) = _ |
| rw [Similarity.minimumArea_map _ _ hr L] |
| change (edge / 2) ^ 2 * TriangularCovering.A_min = _ |
| rw [TriangularCovering.projection_minimum] |
| have hs : Real.sqrt 2 ≠ 0 := ne_of_gt (Real.sqrt_pos.2 (by norm_num)) |
| change (edge / 2) ^ 2 * Real.sqrt 2 = _ |
| field_simp [hs] |
| nlinarith [Real.sq_sqrt (show (0 : ℝ) ≤ 2 by norm_num)] |
|
|
| theorem regular_tetrahedron_volume (v : Fin 4 → Space) |
| (hv : AffineIndependent ℝ v) {edge : ℝ} (hedge : 0 < edge) |
| (hdist : ∀ i j : Fin 4, i ≠ j → dist (v i) (v j) = edge) : |
| volume.real (convexHull ℝ (range v)) = edge ^ 3 / (6 * Real.sqrt 2) := by |
| have hmin := (simplex_chord_projection 3 (by omega) v hv).2.2 |
| change minimumShadowArea (convexHull ℝ (range v)) = |
| 3 * volume.real (convexHull ℝ (range v)) / Metric.diam (convexHull ℝ (range v)) at hmin |
| rw [regular_minimum_shadow_area v hv edge hedge hdist, |
| SimplexDiameter.regular_convexHull_diameter v hedge.le hdist] at hmin |
| have hm := (eq_div_iff hedge.ne').mp hmin |
| calc |
| volume.real (convexHull ℝ (range v)) = |
| (edge ^ 2 / (2 * Real.sqrt 2)) * edge / 3 := by linarith only [hm] |
| _ = edge ^ 3 / (6 * Real.sqrt 2) := by ring |
| |
| theorem regular_edge_shadow_minimum (v : Fin 4 → Space) |
| (hv : AffineIndependent ℝ v) {edge : ℝ} (hedge : 0 < edge) |
| (hdist : ∀ i j : Fin 4, i ≠ j → dist (v i) (v j) = edge) |
| (i j : Fin 4) (hij : i ≠ j) : |
| shadowArea (convexHull ℝ (range v)) (‖v j - v i‖⁻¹ • (v j - v i)) = |
| minimumShadowArea (convexHull ℝ (range v)) := by |
| exact regular_edge_shadow_eq_minimum_of_simplexChordProjection |
| simplex_chord_projection v hv hedge hdist hij |
| |
| theorem regular_edge_shadow_area (v : Fin 4 → Space) |
| (hv : AffineIndependent ℝ v) {edge : ℝ} (hedge : 0 < edge) |
| (hdist : ∀ i j : Fin 4, i ≠ j → dist (v i) (v j) = edge) |
| (i j : Fin 4) (hij : i ≠ j) : |
| shadowArea (convexHull ℝ (range v)) (‖v j - v i‖⁻¹ • (v j - v i)) = |
| edge ^ 2 / (2 * Real.sqrt 2) := |
| (regular_edge_shadow_minimum v hv hedge hdist i j hij).trans |
| (regular_minimum_shadow_area v hv edge hedge hdist) |
| |
| theorem regularMetricNormalization_result : RegularMetricNormalizationStatement := by |
| intro v hv edge hedge hdist |
| refine ⟨SimplexDiameter.regular_convexHull_diameter v hedge.le hdist, |
| regular_tetrahedron_volume v hv hedge hdist, ?_⟩ |
| rw [regular_minimum_shadow_area v hv edge hedge hdist] |
| ring |
| |
| theorem regularProjectionNormalization_result : RegularProjectionNormalizationStatement := |
| regularProjectionNormalization_of_minimum simplex_chord_projection regular_minimum_shadow_area |
| |
| theorem simplexProjectionIdentity_result : SimplexProjectionIdentityStatement := |
| ⟨simplex_chord_projection, tetrahedron_edge_minimizer, |
| regularProjectionNormalization_result⟩ |
| |
| end CylinderCovering |
| |
| end |
| |
| end OAI |
| |
