| import OAI.Geometry.TriangularCovering.Main |
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| namespace OAI |
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| noncomputable section |
| open Set MeasureTheory |
| open scoped InnerProductSpace |
| open TriangularCovering |
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| namespace CubicLowerGeometry |
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| 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)] |
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| 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 |
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| 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)) |
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| 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) |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| end CubicLowerGeometry |
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| end |
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| end OAI |
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