| import OAI.Geometry.CylinderCovering.Caps.FiniteDefinitions |
| import OAI.Geometry.CylinderCovering.Convexity.GlobalDefinitions |
| import OAI.Geometry.CylinderCovering.Physical.Coverage |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set |
| namespace CylinderCovering.HalfheightFinite |
|
|
| theorem height_mem_Icc (hX : |X| ≤ 1 - Z) (hY : |Y| ≤ Z) : |
| Z ∈ Icc (0 : ℝ) 1 := by |
| constructor <;> linarith [abs_nonneg X, abs_nonneg Y] |
|
|
| theorem tetrahedron_eq_unitEdge : tetrahedron = UnitEdgeFootprints.tetrahedron := by |
| ext z |
| constructor |
| · rintro ⟨X, Y, Z, hX, hY, rfl⟩ |
| refine ⟨Z, height_mem_Icc hX hY, rfl, ?_, ?_⟩ |
| · change |X / 2| ≤ (1 - Z) / 2 |
| rw [abs_div, abs_of_pos (by norm_num : (0 : ℝ) < 2)] |
| linarith |
| · change |Y / 2| ≤ Z / 2 |
| rw [abs_div, abs_of_pos (by norm_num : (0 : ℝ) < 2)] |
| linarith |
| · rintro ⟨Z, hZ, hz, hX, hY⟩ |
| refine ⟨2 * z 0, 2 * z 1, Z, ?_, ?_, ?_⟩ |
| · rw [abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 2)] |
| linarith |
| · rw [abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 2)] |
| linarith |
| · ext i |
| fin_cases i |
| · simp [point] |
| · simp [point] |
| · change z 2 = height * Z |
| exact hz |
|
|
| theorem tetrahedron_similarity : tetrahedron = |
| Similarity.map (0 : Space) (1 / 2) (LinearIsometryEquiv.refl ℝ Space) '' |
| TriangularCovering.K := by |
| rw [tetrahedron_eq_unitEdge] |
| exact UnitEdgeFootprints.tetrahedron_similarity |
|
|
| theorem unit_edge_geometry : |
| ∃ v : Fin 4 → Space, AffineIndependent ℝ v ∧ |
| (∀ i j : Fin 4, i ≠ j → dist (v i) (v j) = 1) ∧ |
| tetrahedron = convexHull ℝ (range v) := by |
| let f := AffineMap.homothety (0 : Space) (1 / 2 : ℝ) |
| have hf : Function.Injective f := |
| AffineMap.homothety_injective _ (by norm_num : (1 / 2 : ℝ) ≠ 0) |
| refine ⟨f ∘ TriangularCovering.tetraVertices, |
| TriangularCovering.tetra_affineIndependent.map' f hf, ?_, ?_⟩ |
| · intro i j hij |
| change dist (f (TriangularCovering.tetraVertices i)) |
| (f (TriangularCovering.tetraVertices j)) = 1 |
| simp only [f, AffineMap.homothety_apply, vsub_eq_sub, sub_zero, |
| vadd_eq_add, add_zero] |
| rw [dist_smul₀, Real.norm_eq_abs, |
| TriangularCovering.tetra_edge_length hij] |
| norm_num |
| · have hmap : Similarity.map (0 : Space) (1 / 2) |
| (LinearIsometryEquiv.refl ℝ Space) = f := by |
| funext z |
| simp [Similarity.map, f, AffineMap.homothety_apply] |
| rw [tetrahedron_similarity, TriangularCovering.K_eq_convexHull, |
| hmap, f.image_convexHull, range_comp] |
| |
| theorem isRegularTetrahedron : CylinderCovering.IsRegularTetrahedron tetrahedron := by |
| obtain ⟨v, hv, hd, hK⟩ := unit_edge_geometry |
| exact ⟨v, hv, 1, by norm_num, hd, hK⟩ |
| |
| theorem minimum_area : Similarity.minimumArea tetrahedron = height / 2 := by |
| rw [tetrahedron_eq_unitEdge] |
| exact UnitEdgeFootprints.minimum_area |
| |
| theorem strict_cost_iff_half_minimum (cost : ℝ) : |
| cost < height / 4 ↔ cost < Similarity.minimumArea tetrahedron / 2 := by |
| rw [minimum_area] |
| ring_nf |
| |
| end CylinderCovering.HalfheightFinite |
| |
| end |
| |
| end OAI |
| |
