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785c727 | 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 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | 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
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