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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 | 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 {X Y Z : ℝ} (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
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