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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 | import OAI.Geometry.TriangularCovering.Main
namespace OAI
noncomputable section
open Set MeasureTheory
open scoped InnerProductSpace
open TriangularCovering
namespace CubicLowerGeometry
def vertices (a b T : ℝ) : Fin 3 → Space :=
![point 0 0 (-1), point 0 (a * T) (T - 1), point 0 (b * T) (T - 1)]
lemma triple (a b T ε A B : ℝ) :
inner ℝ (unit (tiltV ε A B))
(cross3 (vertices a b T 1 - vertices a b T 0)
(vertices a b T 2 - vertices a b T 0)) =
H * (a - b) * T ^ 2 / ‖tiltV ε A B‖ := by
simp only [TriangularCovering.unit, inner_smul_left, conj_trivial, inner_coord, cross3_coord]
dsimp [vertices, point, tiltV]
ring
lemma triple_ne_zero {a b T : ℝ} (hab : a < b) (hT : 0 < T)
(ε A B : ℝ) :
inner ℝ (unit (tiltV ε A B))
(cross3 (vertices a b T 1 - vertices a b T 0)
(vertices a b T 2 - vertices a b T 0)) ≠ 0 := by
rw [triple]
exact div_ne_zero (mul_ne_zero (mul_ne_zero H_ne_zero
(sub_ne_zero.mpr (ne_of_lt hab))) (pow_ne_zero _ (ne_of_gt hT)))
(norm_ne_zero_iff.mpr (tiltV_ne_zero ε A B))
def lowerCylinder {a b T : ℝ} (hab : a < b) (hT : 0 < T)
(ε A B : ℝ) : TriCylinder :=
projectedCylinder (unit (tiltV ε A B)) (unit_norm (tiltV_ne_zero ε A B))
(vertices a b T 0) (vertices a b T 1) (vertices a b T 2)
(triple_ne_zero hab hT ε A B)
lemma lowerCylinder_axis {a b T : ℝ} (hab : a < b) (hT : 0 < T)
(ε A B : ℝ) :
(lowerCylinder hab hT ε A B).axis = unit (tiltV ε A B) := rfl
lemma lowerCylinder_area {a b T : ℝ} (hab : a < b) (hT : 0 < T)
(ε A B : ℝ) :
area (lowerCylinder hab hT ε A B).base =
H * (b - a) * T ^ 2 / (2 * Real.sqrt (1 + ε ^ 2 * (A ^ 2 + 2 * B ^ 2))) := by
rw [lowerCylinder, projectedCylinder_area, triple, tiltV_norm]
rw [abs_div, abs_mul, abs_mul, abs_of_pos H_pos,
abs_of_neg (sub_neg.mpr hab), abs_sq, abs_of_nonneg (Real.sqrt_nonneg _)]
ring
lemma triangle_mem {a b T y s : ℝ} (hab : a < b) (hT : 0 < T)
(hs0 : 0 ≤ s) (hsT : s ≤ T) (hay : a * s ≤ y) (hyb : y ≤ b * s) :
point 0 y (s - 1) ∈ triangle (vertices a b T 0)
(vertices a b T 1) (vertices a b T 2) := by
obtain ⟨r, q, hr, hq, hrq, hes, hey⟩ := sector_weights hab hT hs0 hsT hay hyb
rw [mem_triangle_iff]
refine ⟨r, q, hr, hq, hrq, ?_⟩
ext i
fin_cases i <;> dsimp [vertices, point]
· ring
· nlinarith only [hey]
· linear_combination -H * hes
lemma triangle_mem_iff {a b T y s : ℝ} (hab : a < b) (hT : 0 < T) :
point 0 y (s - 1) ∈ triangle (vertices a b T 0)
(vertices a b T 1) (vertices a b T 2) ↔
0 ≤ s ∧ s ≤ T ∧ a * s ≤ y ∧ y ≤ b * s := by
constructor
· intro hm
obtain ⟨r, q, hr, hq, hrq, hv⟩ := (mem_triangle_iff _ _ _ _).mp hm
have hy := congrArg (fun v : Space => v 1) hv
have hs := congrArg (fun v : Space => v 2) hv
dsimp [vertices, point] at hy hs
have hsr : s = r * T + q * T := by
apply mul_left_cancel₀ H_ne_zero
nlinarith only [hs]
refine ⟨?_, ?_, ?_, ?_⟩
· rw [hsr]
positivity
· nlinarith [mul_le_mul_of_nonneg_right hrq hT.le]
· rw [hsr, hy]
nlinarith [mul_nonneg hq (mul_nonneg (sub_nonneg.mpr hab.le) hT.le)]
· rw [hsr, hy]
nlinarith [mul_nonneg hr (mul_nonneg (sub_nonneg.mpr hab.le) hT.le)]
· rintro ⟨hs, hsT, hl, hu⟩
exact triangle_mem hab hT hs hsT hl hu
lemma lowerCylinder_hit {a b T : ℝ} (hab : a < b) (hT : 0 < T)
(ε A B x y z : ℝ)
(hs0 : 0 ≤ 1 + z - ε * x * B) (hsT : 1 + z - ε * x * B ≤ T)
(hl : a * (1 + z - ε * x * B) ≤ y - ε * x * A)
(hu : y - ε * x * A ≤ b * (1 + z - ε * x * B)) :
point x y z ∈ (lowerCylinder hab hT ε A B).carrier := by
apply projectedCylinder_mem_direction (tiltV ε A B) (tiltV_ne_zero ε A B)
_ _ _ _ (triangle_mem hab hT hs0 hsT hl hu) x
ext i
fin_cases i <;> dsimp [point, tiltV] <;> ring
end CubicLowerGeometry
end
end OAI
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