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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 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 | import OAI.Geometry.TriangularCovering.Main
import OAI.Geometry.CylinderCovering.Angular.InterceptDefinitions
namespace OAI
open Set MeasureTheory
open scoped InnerProductSpace
noncomputable section
namespace TriangularCovering
theorem lowerIntercept_eq_convexHull {a b T : ℝ} (hab : a < b) (hT : 0 < T) :
lowerIntercept a b T = convexHull ℝ (Set.range (lowerVertices a b T)) := by
change lowerIntercept a b T =
triangle (lowerVertices a b T 0) (lowerVertices a b T 1) (lowerVertices a b T 2)
ext z
constructor
· rintro ⟨y,t,ht0,htT,hay,hyb,rfl⟩
exact lower_triangle_mem hab hT ht0 htT hay hyb
· rw [mem_triangle_iff]
rintro ⟨r,s,hr,hs,hrs,rfl⟩
refine ⟨r*(a*T)+s*(b*T), r*T+s*T, ?_, ?_, ?_, ?_, ?_⟩
· positivity
· nlinarith [mul_le_mul_of_nonneg_right hrs hT.le]
· nlinarith [mul_le_mul_of_nonneg_left
(mul_le_mul_of_nonneg_right hab.le hT.le) hs]
· nlinarith [mul_le_mul_of_nonneg_left
(mul_le_mul_of_nonneg_right hab.le hT.le) hr]
· ext i
fin_cases i <;> simp [lowerVertices, point]
ring
theorem upperIntercept_eq_convexHull {a b T : ℝ} (hab : a < b) (hT : 0 < T) :
upperIntercept a b T = convexHull ℝ (Set.range (upperVertices a b T)) := by
change upperIntercept a b T =
triangle (upperVertices a b T 0) (upperVertices a b T 1) (upperVertices a b T 2)
ext z
constructor
· rintro ⟨x,t,ht0,htT,hax,hxb,rfl⟩
exact upper_triangle_mem hab hT ht0 htT hax hxb
· rw [mem_triangle_iff]
rintro ⟨r,s,hr,hs,hrs,rfl⟩
refine ⟨r*(a*T)+s*(b*T), r*T+s*T, ?_, ?_, ?_, ?_, ?_⟩
· positivity
· nlinarith [mul_le_mul_of_nonneg_right hrs hT.le]
· nlinarith [mul_le_mul_of_nonneg_left
(mul_le_mul_of_nonneg_right hab.le hT.le) hs]
· nlinarith [mul_le_mul_of_nonneg_left
(mul_le_mul_of_nonneg_right hab.le hT.le) hr]
· ext i
fin_cases i <;> simp [upperVertices, point]
ring
theorem projectedCylinder_carrier_direction (v : Space) (hv : v ≠ 0)
(a b c : Space)
(h : inner ℝ (unit v) (cross3 (b-a) (c-a)) ≠ 0) :
(projectedCylinder (unit v) (unit_norm hv) a b c h).carrier =
{z | ∃ p ∈ triangle a b c, ∃ r : ℝ, z = p + r • v} := by
ext z
constructor
· rintro ⟨q,⟨p,hp,rfl⟩,r,hz⟩
refine ⟨p,hp,(r-inner ℝ (unit v) p) / ‖v‖,?_⟩
calc
z = projection (unit v) p + r • unit v := hz
_ = p + (r-inner ℝ (unit v) p) • unit v := by
unfold projection
module
_ = p + ((r-inner ℝ (unit v) p) / ‖v‖) • v := by
simp only [unit, smul_smul, div_eq_mul_inv]
· rintro ⟨p,hp,r,hz⟩
exact projectedCylinder_mem_direction v hv a b c h hp r hz
theorem lowerCylinder_carrier_intercept {a b T : ℝ} (hab : a < b) (hT : 0 < T)
(ε A B : ℝ) :
(lowerCylinder hab hT ε A B).carrier =
{z | ∃ p ∈ lowerIntercept a b T, ∃ r : ℝ, z = p + r • tiltV ε A B} := by
rw [lowerCylinder, projectedCylinder_carrier_direction _ (tiltV_ne_zero ε A B),
lowerIntercept_eq_convexHull hab hT]
rfl
theorem upperCylinder_carrier_intercept {a b T : ℝ} (hab : a < b) (hT : 0 < T)
(ε A B : ℝ) :
(upperCylinder hab hT ε A B).carrier =
{z | ∃ p ∈ upperIntercept a b T, ∃ r : ℝ, z = p + r • tiltW ε A B} := by
rw [upperCylinder, projectedCylinder_carrier_direction _ (tiltW_ne_zero ε A B),
upperIntercept_eq_convexHull hab hT]
rfl
theorem lowerAngularBase_eq_projection_intercept (ε : ℝ) (he : 0 < ε)
(j : Fin (n ε)) :
lowerAngularBase ε j = projection (lowerAngularAxis ε j) ''
lowerIntercept (mesh (n ε) j) (mesh (n ε) (j + 1))
(cutoff ε (mesh (n ε) j) (mesh (n ε) (j + 1))) := by
rw [lowerIntercept_eq_convexHull (mesh_strict (N := n ε) (ceil_count_pos he) j)
(cutoff_pos_all ε (mesh (n ε) j) (mesh (n ε) (j + 1)))]
rfl
theorem upperAngularBase_eq_projection_intercept (ε : ℝ) (he : 0 < ε)
(j : Fin (n ε)) :
upperAngularBase ε j = projection (upperAngularAxis ε j) ''
upperIntercept (mesh (n ε) j) (mesh (n ε) (j + 1))
(cutoff ε (mesh (n ε) j) (mesh (n ε) (j + 1))) := by
rw [upperIntercept_eq_convexHull (mesh_strict (N := n ε) (ceil_count_pos he) j)
(cutoff_pos_all ε (mesh (n ε) j) (mesh (n ε) (j + 1)))]
rfl
theorem TriCylinder.ext_axis_base {c d : TriCylinder}
(ha : c.axis = d.axis) (hb : c.base = d.base) : c = d := by
cases c
cases d
cases ha
cases hb
rfl
theorem finiteFamily_explicit_laws (ε : ℝ) (he : 0 < ε) :
ExplicitAngularFamily ε (finiteFamily ε he) := by
intro j
simp only [lowerAngularIndex, upperAngularIndex, finiteFamily, Function.comp_apply,
Fin.cast_cast, Fin.cast_eq_self, Fin.append_left, Fin.append_right]
exact ⟨rfl, rfl, rfl, rfl⟩
theorem ExplicitAngularFamily.unique {ε : ℝ}
{c d : Fin (2 * n ε) → TriCylinder}
(hc : ExplicitAngularFamily ε c) (hd : ExplicitAngularFamily ε d) : c = d := by
have h : ∀ k : Fin (n ε + n ε),
c (Fin.cast (two_mul (n ε)).symm k) =
d (Fin.cast (two_mul (n ε)).symm k) := by
intro k
refine Fin.addCases (fun j => ?_) (fun j => ?_) k
· exact TriCylinder.ext_axis_base ((hc j).1.trans (hd j).1.symm)
((hc j).2.1.trans (hd j).2.1.symm)
· exact TriCylinder.ext_axis_base ((hc j).2.2.1.trans (hd j).2.2.1.symm)
((hc j).2.2.2.trans (hd j).2.2.2.symm)
funext i
simpa only [Fin.cast_cast, Fin.cast_eq_self] using h (Fin.cast (two_mul (n ε)) i)
theorem ExplicitAngularFamily.eq_finiteFamily {ε : ℝ} (he : 0 < ε)
{c : Fin (2 * n ε) → TriCylinder} (hc : ExplicitAngularFamily ε c) :
c = finiteFamily ε he := hc.unique (finiteFamily_explicit_laws ε he)
theorem ExplicitAngularFamily.exact_area_sum {ε : ℝ} (he : 0 < ε)
{c : Fin (2 * n ε) → TriCylinder} (hc : ExplicitAngularFamily ε c) :
totalArea c / H =
∑ j : Fin (n ε), normalizedTerm ε (mesh (n ε) j) (mesh (n ε) (j + 1)) := by
rw [hc.eq_finiteFamily he]
exact finiteFamily_total ε he
theorem ExplicitAngularFamily.covers {ε : ℝ} (he : 0 < ε)
(hsmall : ε ≤ 1 / 2000) {c : Fin (2 * n ε) → TriCylinder}
(hc : ExplicitAngularFamily ε c) : Covers c := by
rw [hc.eq_finiteFamily he]
apply finiteFamily_covers he
norm_num [η] at *
exact hsmall
end TriangularCovering
end
end OAI
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