File size: 7,055 Bytes
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 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 | import OAI.Geometry.CylinderCovering.Caps.FiniteDefinitions
import OAI.Geometry.CylinderCovering.Caps.Caps
import OAI.Geometry.CylinderCovering.Footprints.Containment
import OAI.Geometry.CylinderCovering.Radial.Riemann
namespace OAI
open Set MeasureTheory Filter
open scoped Topology InnerProductSpace
noncomputable section
namespace CylinderCovering.HalfheightFinite.CoverageLemmas
lemma firstRaw_ne_zero (δ r : ℝ) : firstRaw δ r ≠ 0 := by
intro h
have h0 := congrArg (fun z : Space => z 0) h
norm_num [firstRaw] at h0
lemma secondRaw_ne_zero (δ r : ℝ) : secondRaw δ r ≠ 0 := by
intro h
have h1 := congrArg (fun z : Space => z 1) h
norm_num [secondRaw] at h1
lemma mem_projected_cylinder_of_intercept {v p z : Space} (hv : v ≠ 0)
{S : Set Space} (hp : p ∈ S) (s : ℝ) (hz : z = p + s • v) :
z ∈ RadialSweep.cylinder (‖v‖⁻¹ • v)
(TriangularCovering.projection (‖v‖⁻¹ • v) '' S) := by
let u := ‖v‖⁻¹ • v
have hn : ‖v‖ ≠ 0 := norm_ne_zero_iff.mpr hv
have hscale : (s * ‖v‖) • u = s • v := by
dsimp [u]
rw [smul_smul, mul_assoc, mul_inv_cancel₀ hn, mul_one]
refine ⟨TriangularCovering.projection u p, ⟨p, hp, rfl⟩,
inner ℝ u p + s * ‖v‖, ?_⟩
rw [hz, TriangularCovering.projection, add_smul, hscale]
module
lemma halfwidth_nonneg (N : ℕ) : 0 ≤ halfwidth N := by
unfold halfwidth
positivity
lemma halfwidth_eq_step (N : ℕ) :
halfwidth N = RadialRiemann.step (-1) 1 N / 2 := by
unfold halfwidth RadialRiemann.step
ring
lemma cell_midpoint_dist {N : ℕ} {i : Fin N} {r : ℝ}
(hr : r ∈ RadialRiemann.cell (-1) 1 N i) :
|r - midpoint N i| ≤ halfwidth N := by
rw [halfwidth_eq_step, abs_le]
have hstep := RadialRiemann.point_succ (-1) 1 N i
unfold midpoint
constructor <;> linarith [hr.1, hr.2]
lemma midpoint_tagged (N : ℕ) : RadialRiemann.IsTagged (-1) 1 N (midpoint N) := by
intro i
have hstep := RadialRiemann.point_succ (-1) 1 N i
have hpos := RadialRiemann.step_pos (by norm_num : (-1:ℝ) < 1)
(Nat.zero_lt_of_lt i.isLt)
change _ ≤ midpoint N i ∧ midpoint N i ≤ _
unfold midpoint
constructor <;> linarith
lemma first_intercept_identity (δ r u X p : ℝ) :
point X (u*r+δ*X*alpha r) (u+δ*X*beta r) =
firstFoot (u+δ*X*(r-p)/2, u*r+δ*X*(r-p)*(r+p)/4) +
(X/2) • firstRaw δ p := by
ext j
fin_cases j <;> simp [point, firstFoot, firstRaw, alpha, beta] <;> ring
lemma second_intercept_identity (δ r u Y p : ℝ) :
point (u*r-δ*Y*alpha r) Y (1-u+δ*Y*beta r) =
secondFoot (u-δ*Y*(r-p)/2, u*r-δ*Y*(r-p)*(r+p)/4) +
(Y/2) • secondRaw δ p := by
ext j
fin_cases j <;> simp [point, secondFoot, secondRaw, alpha, beta] <;> ring
lemma first_intercept_mem {δ : ℝ} (hδ : 0 ≤ δ) {R : ℝ → ℝ}
(hR : ContinuousOn R (Icc (-1:ℝ) 1)) {N : ℕ} (i : Fin N)
{r u X : ℝ} (hr : r ∈ RadialRiemann.cell (-1) 1 N i)
(hu : u ∈ Icc (0:ℝ) (R r)) (hX : |X| ≤ 1) :
(u+δ*X*(r-midpoint N i)/2,
u*r+δ*X*(r-midpoint N i)*(r+midpoint N i)/4) ∈ foot δ R N i := by
have hd := cell_midpoint_dist hr
have he : 0 ≤ collar δ N :=
div_nonneg (mul_nonneg hδ (halfwidth_nonneg N)) (by norm_num)
have hσ := FootprintContainment.halfheight_displacement hδ hX hd
have hM : u ≤ maxCap R N i :=
hu.2.trans (RadialRiemann.le_capMax (by norm_num) hR hr)
have h := FootprintContainment.halfheight_trapezoid
(halfwidth_nonneg N) he hd hu.1 hM hσ
(v := u*r+δ*X*(r-midpoint N i)*(r+midpoint N i)/4)
(by ring) rfl
exact ⟨⟨h.1, h.2.1⟩, h.2.2⟩
lemma second_intercept_mem {δ : ℝ} (hδ : 0 ≤ δ) {R : ℝ → ℝ}
(hR : ContinuousOn R (Icc (-1:ℝ) 1)) {N : ℕ} (i : Fin N)
{r u Y : ℝ} (hr : r ∈ RadialRiemann.cell (-1) 1 N i)
(hu : u ∈ Icc (0:ℝ) (R r)) (hY : |Y| ≤ 1) :
(u-δ*Y*(r-midpoint N i)/2,
u*r-δ*Y*(r-midpoint N i)*(r+midpoint N i)/4) ∈ foot δ R N i := by
have hd := cell_midpoint_dist hr
have he : 0 ≤ collar δ N :=
div_nonneg (mul_nonneg hδ (halfwidth_nonneg N)) (by norm_num)
have hσ : |-(δ*Y*(r-midpoint N i)/2)| ≤ collar δ N := by
rw [abs_neg]
exact FootprintContainment.halfheight_displacement hδ hY hd
have hM : u ≤ maxCap R N i :=
hu.2.trans (RadialRiemann.le_capMax (by norm_num) hR hr)
have h := FootprintContainment.halfheight_trapezoid
(halfwidth_nonneg N) he hd hu.1 hM hσ
(w := u-δ*Y*(r-midpoint N i)/2)
(v := u*r-δ*Y*(r-midpoint N i)*(r+midpoint N i)/4)
(by ring) (by ring)
exact ⟨⟨h.1, h.2.1⟩, h.2.2⟩
lemma first_point_mem_cylinder {δ : ℝ} (hδ : 0 ≤ δ) {R : ℝ → ℝ}
(hR : ContinuousOn R (Icc (-1:ℝ) 1)) {N : ℕ} (i : Fin N)
{r u X : ℝ} (hr : r ∈ RadialRiemann.cell (-1) 1 N i)
(hu : u ∈ Icc (0:ℝ) (R r)) (hX : |X| ≤ 1) :
point X (u*r+δ*X*alpha r) (u+δ*X*beta r) ∈
RadialSweep.cylinder (firstAxis δ (midpoint N i)) (firstBase δ R N i) := by
apply mem_projected_cylinder_of_intercept (firstRaw_ne_zero δ (midpoint N i))
(s := X/2)
· exact ⟨_, first_intercept_mem hδ hR i hr hu hX, rfl⟩
· exact first_intercept_identity δ r u X (midpoint N i)
lemma second_point_mem_cylinder {δ : ℝ} (hδ : 0 ≤ δ) {R : ℝ → ℝ}
(hR : ContinuousOn R (Icc (-1:ℝ) 1)) {N : ℕ} (i : Fin N)
{r u Y : ℝ} (hr : r ∈ RadialRiemann.cell (-1) 1 N i)
(hu : u ∈ Icc (0:ℝ) (R r)) (hY : |Y| ≤ 1) :
point (u*r-δ*Y*alpha r) Y (1-u+δ*Y*beta r) ∈
RadialSweep.cylinder (secondAxis δ (midpoint N i)) (secondBase δ R N i) := by
apply mem_projected_cylinder_of_intercept (secondRaw_ne_zero δ (midpoint N i))
(s := Y/2)
· exact ⟨_, second_intercept_mem hδ hR i hr hu hY, rfl⟩
· exact second_intercept_identity δ r u Y (midpoint N i)
theorem family_covers_of_covered {δ : ℝ} (hδ : 0 ≤ δ) {R : ℝ → ℝ}
(hR : ContinuousOn R (Icc (-1:ℝ) 1)) (hcover : HalfHeightCaps.Covered δ R)
{N : ℕ} (hN : 0 < N) :
∀ z ∈ tetrahedron,
(∃ i : Fin N, z ∈ RadialSweep.cylinder
(firstAxis δ (midpoint N i)) (firstBase δ R N i)) ∨
(∃ i : Fin N, z ∈ RadialSweep.cylinder
(secondAxis δ (midpoint N i)) (secondBase δ R N i)) := by
rintro z ⟨X,Y,Z,hX,hY,rfl⟩
have hX1 : |X| ≤ 1 := by linarith [abs_nonneg Y]
have hY1 : |Y| ≤ 1 := by linarith [abs_nonneg X]
rcases hcover X Y Z hX hY with ⟨r,u,hr,hu,hRu,hYr,hZr⟩ |
⟨r,u,hr,hu,hRu,hXr,hZr⟩
· obtain ⟨i,hi⟩ := RadialRiemann.exists_cell hN (abs_le.mp hr)
left
refine ⟨i, ?_⟩
rw [hYr,hZr]
exact first_point_mem_cylinder hδ hR i hi ⟨hu,hRu⟩ hX1
· obtain ⟨i,hi⟩ := RadialRiemann.exists_cell hN (abs_le.mp hr)
right
refine ⟨i, ?_⟩
have hZ : Z = 1-u+δ*Y*HalfHeightCaps.beta r := by linarith only [hZr]
rw [hXr,hZ]
exact second_point_mem_cylinder hδ hR i hi ⟨hu,hRu⟩ hY1
end CylinderCovering.HalfheightFinite.CoverageLemmas
end
end OAI
|