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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 | import OAI.Geometry.TriangularCovering.Main
import OAI.Geometry.CylinderCovering.Radial.Triangle
namespace OAI
noncomputable section
namespace RadialProjection
open TriangularCovering Set MeasureTheory
open scoped InnerProductSpace
lemma cross3_cross_right (u v w : Space) :
cross3 u (cross3 v w) = (inner ℝ u w) • v - (inner ℝ u v) • w := by
ext i
fin_cases i <;> simp [cross3_coord, inner_coord] <;> ring
lemma cross3_normal {e a b : Space} (he : ‖e‖ = 1)
(ha : inner ℝ e a = 0) (hb : inner ℝ e b = 0) :
cross3 a b = (inner ℝ e (cross3 a b)) • e := by
have hc : cross3 e (cross3 a b) = 0 := by
rw [cross3_cross_right, ha, hb]
simp
have hh := cross3_cross_right e e (cross3 a b)
rw [hc] at hh
have hz : cross3 e 0 = 0 := by
ext i
fin_cases i <;> simp [cross3_coord]
rw [hz, real_inner_self_eq_norm_sq, he, one_pow, one_smul] at hh
exact (sub_eq_zero.mp hh.symm).symm
lemma sideways_norm {e w : Space} (he : ‖e‖ = 1)
(hw : inner ℝ e w = 0) : ‖e + w‖ = Real.sqrt (1 + ‖w‖ ^ 2) := by
have hsq : ‖e + w‖ ^ 2 = 1 + ‖w‖ ^ 2 := by
rw [norm_add_sq_real, he, hw]
ring
rw [← hsq, Real.sqrt_sq (norm_nonneg _)]
lemma sideways_ne_zero {e w : Space} (he : ‖e‖ = 1)
(hw : inner ℝ e w = 0) : e + w ≠ 0 := by
intro h
have hn := sideways_norm he hw
rw [h, norm_zero] at hn
have hp : 0 < Real.sqrt (1 + ‖w‖ ^ 2) := by positivity
linarith
lemma normal_cross_abs {e a b : Space} (he : ‖e‖ = 1)
(ha : inner ℝ e a = 0) (hb : inner ℝ e b = 0) :
|inner ℝ e (cross3 a b)| =
Real.sqrt (inner ℝ a a * inner ℝ b b - (inner ℝ a b) ^ 2) := by
have hh := triple_gram_identity e a b
rw [ha, hb, real_inner_self_eq_norm_sq e, he] at hh
norm_num at hh
rw [real_inner_self_eq_norm_sq a, real_inner_self_eq_norm_sq b,
← hh, Real.sqrt_sq_eq_abs]
theorem normal_projection_factor {e a b w : Space} (he : ‖e‖ = 1)
(ha : inner ℝ e a = 0) (hb : inner ℝ e b = 0)
(hw : inner ℝ e w = 0) :
|inner ℝ (unit (e + w)) (cross3 a b)| =
Real.sqrt (inner ℝ a a * inner ℝ b b - (inner ℝ a b) ^ 2) /
Real.sqrt (1 + ‖w‖ ^ 2) := by
have hcrossw : inner ℝ w (cross3 a b) = 0 := by
rw [cross3_normal he ha hb, inner_smul_right, real_inner_comm e w, hw]
simp
rw [TriangularCovering.unit, inner_smul_left, inner_add_left, hcrossw, add_zero]
simp only [conj_trivial, abs_mul, abs_inv, abs_norm]
rw [normal_cross_abs he ha hb, sideways_norm he hw]
ring
lemma cross3_coordinates (a b : Space) (u₁ v₁ u₂ v₂ : ℝ) :
cross3 (u₁ • a + v₁ • b) (u₂ • a + v₂ • b) =
(u₁ * v₂ - v₁ * u₂) • cross3 a b := by
ext i
fin_cases i <;> simp [cross3_coord] <;> ring
lemma collar_cross (p₀ a b : Space) (l r m d : ℝ) :
cross3
((p₀ + r • a + (m * r - d / 2 * (r - l)) • b) -
(p₀ + l • a + (m * l) • b))
((p₀ + r • a + (m * r + d / 2 * (r - l)) • b) -
(p₀ + l • a + (m * l) • b)) =
(d * (r - l) ^ 2) • cross3 a b := by
have h₁ : (p₀ + r • a + (m * r - d / 2 * (r - l)) • b) -
(p₀ + l • a + (m * l) • b) =
(r - l) • a + (m * (r - l) - d / 2 * (r - l)) • b := by module
have h₂ : (p₀ + r • a + (m * r + d / 2 * (r - l)) • b) -
(p₀ + l • a + (m * l) • b) =
(r - l) • a + (m * (r - l) + d / 2 * (r - l)) • b := by module
rw [h₁, h₂, cross3_coordinates]
congr 1
ring
theorem collar_projection_area (p₀ e a b w : Space) (l r m d : ℝ)
(he : ‖e‖ = 1) (ha : inner ℝ e a = 0) (hb : inner ℝ e b = 0)
(hw : inner ℝ e w = 0) (hd : 0 ≤ d) :
area (projection (TriangularCovering.unit (e + w)) '' triangle
(p₀ + l • a + (m * l) • b)
(p₀ + r • a + (m * r - d / 2 * (r - l)) • b)
(p₀ + r • a + (m * r + d / 2 * (r - l)) • b)) =
Real.sqrt (inner ℝ a a * inner ℝ b b - (inner ℝ a b) ^ 2) *
d * (r - l) ^ 2 / (2 * Real.sqrt (1 + ‖w‖ ^ 2)) := by
unfold area
rw [projection_triangle_measure (unit_norm (sideways_ne_zero he hw)),
ENNReal.toReal_ofReal (by positivity), collar_cross, inner_smul_right,
abs_mul, abs_of_nonneg (mul_nonneg hd (sq_nonneg _)),
normal_projection_factor he ha hb hw]
ring
theorem collar_projection_nondegenerate (p₀ e a b w : Space) (l r m d : ℝ)
(he : ‖e‖ = 1) (ha : inner ℝ e a = 0) (hb : inner ℝ e b = 0)
(hw : inner ℝ e w = 0) (hd : 0 < d) (hlr : l < r)
(hκ : 0 < Real.sqrt (inner ℝ a a * inner ℝ b b - (inner ℝ a b) ^ 2)) :
inner ℝ (TriangularCovering.unit (e + w))
(cross3
((p₀ + r • a + (m * r - d / 2 * (r - l)) • b) -
(p₀ + l • a + (m * l) • b))
((p₀ + r • a + (m * r + d / 2 * (r - l)) • b) -
(p₀ + l • a + (m * l) • b))) ≠ 0 := by
rw [← abs_pos, collar_cross, inner_smul_right, abs_mul,
abs_of_nonneg (mul_nonneg hd.le (sq_nonneg _)), normal_projection_factor he ha hb hw]
exact mul_pos (mul_pos hd (sq_pos_of_pos (sub_pos.mpr hlr))) (div_pos hκ (by positivity))
lemma gram_sqrt_pos {a b : Space}
(hab : LinearIndependent ℝ (![a, b] : Fin 2 → Space)) :
0 < Real.sqrt (inner ℝ a a * inner ℝ b b - (inner ℝ a b) ^ 2) := by
have hpi : LinearIndependent ℝ (![a.ofLp, b.ofLp] : Fin 2 → (Fin 3 → ℝ)) := by
have hh := hab.map' (EuclideanSpace.equiv (Fin 3) ℝ).toLinearMap
(LinearMap.ker_eq_bot.mpr (EuclideanSpace.equiv (Fin 3) ℝ).injective)
have hfun : (EuclideanSpace.equiv (Fin 3) ℝ).toLinearMap ∘
(![a, b] : Fin 2 → Space) = ![a.ofLp, b.ofLp] := by
funext i
fin_cases i <;> rfl
rw [hfun] at hh
exact hh
have hc : cross3 a b ≠ 0 := by
intro hz
apply (crossProduct_ne_zero_iff_linearIndependent.mpr hpi)
simpa [cross3] using congrArg WithLp.ofLp hz
rw [← cross3_gram, real_inner_self_eq_norm_sq, Real.sqrt_sq (norm_nonneg _)]
exact norm_pos_iff.mpr hc
def coordinateAffine (p₀ a b : Space) : (ℝ × ℝ) →ᵃ[ℝ] Space where
toFun := fun p => p₀ + p.1 • a + p.2 • b
linear := (LinearMap.fst ℝ ℝ ℝ).smulRight a + (LinearMap.snd ℝ ℝ ℝ).smulRight b
map_vadd' := by
intro p v
simp only [vadd_eq_add, Prod.fst_add, Prod.snd_add, LinearMap.add_apply,
LinearMap.smulRight_apply, LinearMap.fst_apply, LinearMap.snd_apply]
module
lemma coordinate_region_image (p₀ a b : Space) {l r m d : ℝ}
(hlr : l < r) (hd : 0 < d) :
coordinateAffine p₀ a b '' CylinderCovering.RadialTriangle.region l r m d =
triangle (p₀ + l • a + (m * l) • b)
(p₀ + r • a + (m * r - d / 2 * (r - l)) • b)
(p₀ + r • a + (m * r + d / 2 * (r - l)) • b) := by
rw [CylinderCovering.RadialTriangle.region_eq_convexHull hlr hd,
(coordinateAffine p₀ a b).image_convexHull, triangle]
congr 1
rw [← Set.range_comp]
congr 1
funext i
fin_cases i <;> rfl
theorem exists_collar_cylinder_exact (p₀ e a b w : Space) (l r m d : ℝ)
(he : ‖e‖ = 1) (ha : inner ℝ e a = 0) (hb : inner ℝ e b = 0)
(hw : inner ℝ e w = 0) (hd : 0 < d) (hlr : l < r)
(hab : LinearIndependent ℝ (![a, b] : Fin 2 → Space)) :
∃ c : TriCylinder,
c.axis = TriangularCovering.unit (e + w) ∧
c.base = projection (TriangularCovering.unit (e + w)) '' triangle
(p₀ + l • a + (m * l) • b)
(p₀ + r • a + (m * r - d / 2 * (r - l)) • b)
(p₀ + r • a + (m * r + d / 2 * (r - l)) • b) ∧
area c.base = Real.sqrt (inner ℝ a a * inner ℝ b b - (inner ℝ a b) ^ 2) *
d * (r - l) ^ 2 / (2 * Real.sqrt (1 + ‖w‖ ^ 2)) ∧
∀ u v s : ℝ, l ≤ u → u ≤ r → |v - m * u| ≤ d / 2 * (u - l) →
p₀ + u • a + v • b + s • (e + w) ∈ c.carrier := by
let P := p₀ + l • a + (m * l) • b
let Q := p₀ + r • a + (m * r - d / 2 * (r - l)) • b
let R := p₀ + r • a + (m * r + d / 2 * (r - l)) • b
have hv := sideways_ne_zero he hw
have hcross := collar_projection_nondegenerate p₀ e a b w l r m d
he ha hb hw hd hlr (gram_sqrt_pos hab)
refine ⟨projectedCylinder (TriangularCovering.unit (e + w)) (unit_norm hv)
P Q R hcross, rfl, rfl, ?_, ?_⟩
· exact collar_projection_area p₀ e a b w l r m d he ha hb hw hd.le
· intro u v s hlu hur huv
have hp : p₀ + u • a + v • b ∈ triangle P Q R := by
rw [← coordinate_region_image p₀ a b hlr hd]
exact ⟨(u, v), ⟨hlu, hur, huv⟩, rfl⟩
exact projectedCylinder_mem_direction (e + w) hv P Q R hcross hp s rfl
theorem exists_collar_cylinder (p₀ e a b w : Space) (l r m d : ℝ)
(he : ‖e‖ = 1) (ha : inner ℝ e a = 0) (hb : inner ℝ e b = 0)
(hw : inner ℝ e w = 0) (hd : 0 < d) (hlr : l < r)
(hab : LinearIndependent ℝ (![a, b] : Fin 2 → Space)) :
∃ c : TriCylinder,
c.axis = TriangularCovering.unit (e + w) ∧
area c.base = Real.sqrt (inner ℝ a a * inner ℝ b b - (inner ℝ a b) ^ 2) *
d * (r - l) ^ 2 / (2 * Real.sqrt (1 + ‖w‖ ^ 2)) ∧
∀ u v s : ℝ, l ≤ u → u ≤ r → |v - m * u| ≤ d / 2 * (u - l) →
p₀ + u • a + v • b + s • (e + w) ∈ c.carrier := by
obtain ⟨c, haxis, _, harea, hcover⟩ :=
exists_collar_cylinder_exact p₀ e a b w l r m d he ha hb hw hd hlr hab
exact ⟨c, haxis, harea, hcover⟩
end RadialProjection
end
end OAI
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