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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 | import OAI.Geometry.CylinderCovering.Caps.TrapezoidGeometryDefinitions
import OAI.Geometry.CylinderCovering.Caps.FiniteShape
import OAI.Geometry.CylinderCovering.Caps.Geometry
import OAI.Geometry.CylinderCovering.Caps.FiniteTheorem
namespace OAI
open Set Filter
open scoped Topology
noncomputable section
namespace CylinderCovering.HalfheightFinite
theorem trapezoid_pair {δ : ℝ} (hδ : 0 < δ) {R : ℝ → ℝ}
(hR : ContinuousOn R (Icc (-1 : ℝ) 1))
(hR0 : ∀ r ∈ Icc (-1 : ℝ) 1, 0 ≤ R r) {N : ℕ} (i : Fin N) :
NondegenerateTrapezoid (firstBase δ R N i) ∧
NondegenerateTrapezoid (secondBase δ R N i) := by
have hM := RadialRiemann.capMax_nonneg (by norm_num : (-1 : ℝ) < 1) hR hR0 i
obtain ⟨hU, hL, hiU, hiL⟩ := Shape.projected_shape i hδ hM
obtain ⟨hnU, hnL⟩ := Shape.projected_nondegenerate i hδ hM
obtain ⟨ha, hb, vU, vL, hvU, hvL, hU0, hU1, hL0, hL1⟩ :=
Shape.projected_parallel_sides i hδ hM
constructor
· exact ⟨Shape.firstVertices δ R N i, hU, hiU, hnU,
2 * halfwidth N * collar δ N,
2 * halfwidth N * (maxCap R N i + 3 * collar δ N),
ha, hb, vU, hvU, hU0, hU1⟩
· exact ⟨Shape.secondVertices δ R N i, hL, hiL, hnL,
2 * halfwidth N * collar δ N,
2 * halfwidth N * (maxCap R N i + 3 * collar δ N),
ha, hb, vL, hvL, hL0, hL1⟩
theorem source_trapezoids (δ : ℝ) (hδ : 0 < δ) (N : ℕ) (i : Fin N) :
NondegenerateTrapezoid (firstBase δ (cubicCap δ) N i) ∧
NondegenerateTrapezoid (secondBase δ (cubicCap δ) N i) ∧
NondegenerateTrapezoid (firstBase δ (fractionalCap δ) N i) ∧
NondegenerateTrapezoid (secondBase δ (fractionalCap δ) N i) := by
obtain ⟨hcU, hcL⟩ := trapezoid_pair hδ (cubicCap_continuous δ).continuousOn
(fun r _ => (cubicCap_positive hδ.le r).le) i
obtain ⟨hfU, hfL⟩ := trapezoid_pair hδ (fractionalCap_continuous δ).continuousOn
(fun r _ => (fractionalCap_positive hδ.le r).le) i
exact ⟨hcU, hcL, hfU, hfL⟩
theorem eventual_half_minimum : ∀ᶠ δ : ℝ in 𝓝[>] 0,
(∀ᶠ N in atTop, 0 < N ∧
totalArea δ (cubicCap δ) N < Similarity.minimumArea tetrahedron / 2) ∧
(∀ᶠ N in atTop, 0 < N ∧
totalArea δ (fractionalCap δ) N < Similarity.minimumArea tetrahedron / 2) := by
have hthreshold : Similarity.minimumArea tetrahedron / 2 = height / 4 := by
rw [minimum_area]
ring
simpa only [hthreshold] using full_halfheight.2.2.2.2
theorem full_halfheight_geometry : GeometryStatement :=
⟨unit_edge_geometry, minimum_area, source_trapezoids, eventual_half_minimum⟩
end CylinderCovering.HalfheightFinite
end
end OAI
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