| import OAI.Geometry.CylinderCovering.Simplex.Projection | |
| namespace OAI | |
| noncomputable section | |
| open Set MeasureTheory | |
| namespace CylinderCovering | |
| theorem simplex_homothety_volume {d : ℕ} (q : Euclidean d) {r : ℝ} | |
| (hr : 0 ≤ r) (S : Set (Euclidean d)) : | |
| volume.real (AffineMap.homothety q r '' S) = r ^ d * volume.real S := by | |
| change (volume (AffineMap.homothety q r '' S)).toReal = _ | |
| rw [Measure.addHaar_image_homothety] | |
| simp [ENNReal.toReal_mul, abs_of_nonneg (pow_nonneg hr d), | |
| ENNReal.toReal_ofReal (pow_nonneg hr d), Measure.real] | |
| theorem affine_simplex_overlap_volume {d : ℕ} | |
| (p : AffineBasis (Fin (d + 1)) ℝ (Euclidean d)) {u : Euclidean d} | |
| (hu : u ≠ 0) {s : ℝ} (hs : 0 ≤ s) | |
| (hsm : s * (∑ i, max 0 (-(p.coord i).linear u)) < 1) : | |
| volume.real {x | x ∈ convexHull ℝ (range p) ∧ | |
| x + s • u ∈ convexHull ℝ (range p)} = | |
| (1 - s * (∑ i, max 0 (-(p.coord i).linear u))) ^ d * | |
| volume.real (convexHull ℝ (range p)) := by | |
| obtain ⟨q, _, _, hhom⟩ := SimplexChord.affine_intersection_homothety p hu | |
| rw [hhom s hs hsm] | |
| exact simplex_homothety_volume q (sub_nonneg.mpr hsm.le) _ | |
| theorem affine_simplex_overlap_volume_lt_reciprocal {d : ℕ} | |
| (p : AffineBasis (Fin (d + 1)) ℝ (Euclidean d)) {u : Euclidean d} | |
| (hu : u ≠ 0) {s : ℝ} (hs : 0 ≤ s) | |
| (hsM : s < (∑ i, max 0 (-(p.coord i).linear u))⁻¹) : | |
| volume.real {x | x ∈ convexHull ℝ (range p) ∧ | |
| x + s • u ∈ convexHull ℝ (range p)} = | |
| (1 - s * (∑ i, max 0 (-(p.coord i).linear u))) ^ d * | |
| volume.real (convexHull ℝ (range p)) := by | |
| have hM : 0 < ∑ i, max 0 (-(p.coord i).linear u) := | |
| SimplexChord.negative_mass_pos _ (SimplexChord.sum_coord_linear_eq_zero p u) | |
| (SimplexChord.coord_linear_ne_zero p hu) | |
| apply affine_simplex_overlap_volume p hu hs | |
| exact (lt_div_iff₀ hM).mp (by simpa only [one_div] using hsM) | |
| theorem translated_intersection_eq {d : ℕ} (S : Set (Euclidean d)) | |
| (u : Euclidean d) (s : ℝ) : | |
| {x | x ∈ S ∧ x + s • u ∈ S} = S ∩ (fun y => y - s • u) '' S := by | |
| ext x | |
| constructor | |
| · rintro ⟨hx, hxu⟩ | |
| exact ⟨hx, x + s • u, hxu, add_sub_cancel_right x (s • u)⟩ | |
| · rintro ⟨hx, y, hy, hxy⟩ | |
| refine ⟨hx, ?_⟩ | |
| rw [← hxy] | |
| simpa using hy | |
| theorem simplex_overlap_volume {d : ℕ} (v : Fin (d + 1) → Euclidean d) | |
| (hv : AffineIndependent ℝ v) {u : Euclidean d} (hu : ‖u‖ = 1) | |
| {s : ℝ} (hs : 0 ≤ s) (hsL : s < longestChord (simplexCarrier v) u) : | |
| volume.real {x | x ∈ simplexCarrier v ∧ x + s • u ∈ simplexCarrier v} = | |
| (1 - s / longestChord (simplexCarrier v) u) ^ d * | |
| volume.real (simplexCarrier v) := by | |
| obtain ⟨hL, _, _, _, hhom⟩ := simplex_chord_data v hv hu | |
| rw [hhom s hs hsL] | |
| apply simplex_homothety_volume | |
| exact sub_nonneg.mpr ((div_le_one hL).mpr hsL.le) | |
| theorem tetrahedron_overlap_volume (v : Fin 4 → Euclidean 3) | |
| (hv : AffineIndependent ℝ v) {u : Euclidean 3} (hu : ‖u‖ = 1) | |
| {s : ℝ} (hs : 0 ≤ s) (hsL : s < longestChord (simplexCarrier v) u) : | |
| volume.real {x | x ∈ simplexCarrier v ∧ x + s • u ∈ simplexCarrier v} = | |
| (1 - s / longestChord (simplexCarrier v) u) ^ 3 * | |
| volume.real (simplexCarrier v) := | |
| simplex_overlap_volume v hv hu hs hsL | |
| theorem tetrahedron_translated_overlap_volume (v : Fin 4 → Euclidean 3) | |
| (hv : AffineIndependent ℝ v) {u : Euclidean 3} (hu : ‖u‖ = 1) | |
| {s : ℝ} (hs : 0 ≤ s) (hsL : s < longestChord (simplexCarrier v) u) : | |
| volume.real (simplexCarrier v ∩ (fun y => y - s • u) '' simplexCarrier v) = | |
| (1 - s / longestChord (simplexCarrier v) u) ^ 3 * | |
| volume.real (simplexCarrier v) := by | |
| rw [← translated_intersection_eq] | |
| exact tetrahedron_overlap_volume v hv hu hs hsL | |
| end CylinderCovering | |
| end | |
| end OAI | |