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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 | import OAI.Geometry.CylinderCovering.Barycentric.SweepsDefinitions
import OAI.Geometry.CylinderCovering.Radial.Coordinates
namespace OAI
namespace RadialCoordinates
open Set
private theorem barycentric_cap_pos (k t : ℝ) : 0 < cap k t := by
unfold cap capPolynomial
positivity
private theorem firstSweepPoint_first (k t S p : ℝ) :
firstSweepPoint k t S p 0 = S * t + k * p * t ^ 2 := by
simp [firstSweepPoint, barycentricPoint, angular]
ring
private theorem firstSweepPoint_second (k t S p : ℝ) :
firstSweepPoint k t S p 1 = S * (1 - t) - k * p * (1 - t) ^ 2 := by
simp [firstSweepPoint, barycentricPoint, angular]
ring
private theorem firstSweepPoint_difference (k t S p : ℝ) :
firstSweepPoint k t S p 2 - firstSweepPoint k t S p 3 = p := by
simp [firstSweepPoint, barycentricPoint]
ring
private theorem firstSweepPoint_eq {v : BarycentricSpace} {k t S : ℝ}
(hsum : ∑ i, v i = 1)
(ha : v 0 = S * t + k * (v 2 - v 3) * t ^ 2)
(hb : v 1 = S * (1 - t) - k * (v 2 - v 3) * (1 - t) ^ 2) :
firstSweepPoint k t S (v 2 - v 3) = v := by
have heq := radial_representation_sum_difference ha hb
unfold firstSweepPoint
rw [← heq.1, ← heq.2]
exact barycentricPoint_reconstruct hsum
theorem firstSweep_eq_image (k : ℝ) :
firstSweep k = (fun x : ℝ × ℝ × ℝ =>
firstSweepPoint k x.1 (x.2.1 * cap k x.1) x.2.2) ''
(Icc 0 1 ×ˢ (Icc 0 1 ×ˢ Icc (-1) 1)) := by
ext v
constructor
· rintro ⟨hsum, t, S, ht, hS, hcap, hp, ha, hb⟩
have hc := barycentric_cap_pos k t
refine ⟨(t, S / cap k t, v 2 - v 3), ?_, ?_⟩
· exact ⟨ht, ⟨div_nonneg hS (le_of_lt hc), (div_le_one hc).2 hcap⟩,
abs_le.mp hp⟩
· simp only [div_mul_cancel₀ S (ne_of_gt hc)]
exact firstSweepPoint_eq hsum ha hb
· rintro ⟨⟨t, u, p⟩, ⟨ht, hu, hp⟩, rfl⟩
have hc := barycentric_cap_pos k t
refine ⟨barycentricPoint_sum _ _ _, t, u * cap k t, ht,
mul_nonneg hu.1 (le_of_lt hc), ?_, ?_, ?_, ?_⟩
· exact mul_le_of_le_one_left (le_of_lt hc) hu.2
· rw [firstSweepPoint_difference]
exact abs_le.mpr hp
· rw [firstSweepPoint_difference, firstSweepPoint_first]
· rw [firstSweepPoint_difference, firstSweepPoint_second]
theorem firstSweep_isCompact (k : ℝ) : IsCompact (firstSweep k) := by
rw [firstSweep_eq_image]
apply (isCompact_Icc.prod (isCompact_Icc.prod isCompact_Icc)).image
unfold firstSweepPoint barycentricPoint cap capPolynomial angular
fun_prop
private theorem secondSweepPoint_first (k r T q : ℝ) :
secondSweepPoint k r T q 2 = T * r - k * q * r ^ 2 := by
simp [secondSweepPoint, barycentricPoint, angular]
ring
private theorem secondSweepPoint_second (k r T q : ℝ) :
secondSweepPoint k r T q 3 = T * (1 - r) + k * q * (1 - r) ^ 2 := by
simp [secondSweepPoint, barycentricPoint, angular]
ring
private theorem secondSweepPoint_difference (k r T q : ℝ) :
secondSweepPoint k r T q 0 - secondSweepPoint k r T q 1 = q := by
simp [secondSweepPoint, barycentricPoint]
ring
private theorem secondSweepPoint_eq {v : BarycentricSpace} {k r T : ℝ}
(hsum : ∑ i, v i = 1)
(hc : v 2 = T * r - k * (v 0 - v 1) * r ^ 2)
(hd : v 3 = T * (1 - r) + k * (v 0 - v 1) * (1 - r) ^ 2) :
secondSweepPoint k r T (v 0 - v 1) = v := by
have hsum' : v 0 + v 1 + v 2 + v 3 = 1 := by
simpa [Fin.sum_univ_succ, add_assoc] using hsum
have hcd : v 2 + v 3 = T - k * (v 0 - v 1) * angular r := by
unfold angular
nlinarith only [hc, hd]
have hdiff : v 2 - v 3 = T * angular r - k * (v 0 - v 1) *
(1 + angular r ^ 2) / 2 := by
unfold angular
nlinarith only [hc, hd]
have hab : v 0 + v 1 = 1 - T + k * (v 0 - v 1) * angular r := by
linarith only [hsum', hcd]
unfold secondSweepPoint
rw [← hab, ← hdiff]
exact barycentricPoint_reconstruct hsum
theorem secondSweep_eq_image (k : ℝ) :
secondSweep k = (fun x : ℝ × ℝ × ℝ =>
secondSweepPoint k x.1 (x.2.1 * cap k x.1) x.2.2) ''
(Icc 0 1 ×ˢ (Icc 0 1 ×ˢ Icc (-1) 1)) := by
ext v
constructor
· rintro ⟨hsum, r, T, hr, hT, hcap, hq, hc, hd⟩
have hcap0 := barycentric_cap_pos k r
refine ⟨(r, T / cap k r, v 0 - v 1), ?_, ?_⟩
· exact ⟨hr, ⟨div_nonneg hT (le_of_lt hcap0), (div_le_one hcap0).2 hcap⟩,
abs_le.mp hq⟩
· simp only [div_mul_cancel₀ T (ne_of_gt hcap0)]
exact secondSweepPoint_eq hsum hc hd
· rintro ⟨⟨r, u, q⟩, ⟨hr, hu, hq⟩, rfl⟩
have hcap0 := barycentric_cap_pos k r
refine ⟨barycentricPoint_sum _ _ _, r, u * cap k r, hr,
mul_nonneg hu.1 (le_of_lt hcap0), ?_, ?_, ?_, ?_⟩
· exact mul_le_of_le_one_left (le_of_lt hcap0) hu.2
· rw [secondSweepPoint_difference]
exact abs_le.mpr hq
· rw [secondSweepPoint_difference, secondSweepPoint_first]
· rw [secondSweepPoint_difference, secondSweepPoint_second]
theorem secondSweep_isCompact (k : ℝ) : IsCompact (secondSweep k) := by
rw [secondSweep_eq_image]
apply (isCompact_Icc.prod (isCompact_Icc.prod isCompact_Icc)).image
unfold secondSweepPoint barycentricPoint cap capPolynomial angular
fun_prop
theorem firstSweep_isClosed (k : ℝ) : IsClosed (firstSweep k) :=
(firstSweep_isCompact k).isClosed
theorem secondSweep_isClosed (k : ℝ) : IsClosed (secondSweep k) :=
(secondSweep_isCompact k).isClosed
theorem firstSweep_eq_withCap (k : ℝ) : firstSweep k = firstSweepWithCap k (cap k) := by
ext v
constructor
· rintro ⟨hsum, t, S, ht, hS, hcap, hp, ha, hb⟩
exact ⟨((t, S), v 2 - v 3), ⟨⟨ht, hS, hcap⟩, abs_le.mp hp⟩,
firstSweepPoint_eq hsum ha hb⟩
· rintro ⟨⟨⟨t, S⟩, p⟩, ⟨⟨ht, hS, hcap⟩, hp⟩, rfl⟩
refine ⟨barycentricPoint_sum _ _ _, t, S, ht, hS, hcap, ?_, ?_, ?_⟩
· rw [firstSweepPoint_difference]; exact abs_le.mpr hp
· rw [firstSweepPoint_difference, firstSweepPoint_first]
· rw [firstSweepPoint_difference, firstSweepPoint_second]
theorem secondSweep_eq_withCap (k : ℝ) : secondSweep k = secondSweepWithCap k (cap k) := by
ext v
constructor
· rintro ⟨hsum, r, T, hr, hT, hcap, hq, hc, hd⟩
exact ⟨((r, T), v 0 - v 1), ⟨⟨hr, hT, hcap⟩, abs_le.mp hq⟩,
secondSweepPoint_eq hsum hc hd⟩
· rintro ⟨⟨⟨r, T⟩, q⟩, ⟨⟨hr, hT, hcap⟩, hq⟩, rfl⟩
refine ⟨barycentricPoint_sum _ _ _, r, T, hr, hT, hcap, ?_, ?_, ?_⟩
· rw [secondSweepPoint_difference]; exact abs_le.mpr hq
· rw [secondSweepPoint_difference, secondSweepPoint_first]
· rw [secondSweepPoint_difference, secondSweepPoint_second]
end RadialCoordinates
end OAI
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