| import OAI.Geometry.TriangularCovering.Main |
| import OAI.Geometry.CylinderCovering.Angular.InterceptDefinitions |
|
|
| namespace OAI |
|
|
| open Set MeasureTheory |
| open scoped InnerProductSpace |
| noncomputable section |
| namespace TriangularCovering |
|
|
| theorem lowerIntercept_eq_convexHull {a b T : ℝ} (hab : a < b) (hT : 0 < T) : |
| lowerIntercept a b T = convexHull ℝ (Set.range (lowerVertices a b T)) := by |
| change lowerIntercept a b T = |
| triangle (lowerVertices a b T 0) (lowerVertices a b T 1) (lowerVertices a b T 2) |
| ext z |
| constructor |
| · rintro ⟨y,t,ht0,htT,hay,hyb,rfl⟩ |
| exact lower_triangle_mem hab hT ht0 htT hay hyb |
| · rw [mem_triangle_iff] |
| rintro ⟨r,s,hr,hs,hrs,rfl⟩ |
| refine ⟨r*(a*T)+s*(b*T), r*T+s*T, ?_, ?_, ?_, ?_, ?_⟩ |
| · positivity |
| · nlinarith [mul_le_mul_of_nonneg_right hrs hT.le] |
| · nlinarith [mul_le_mul_of_nonneg_left |
| (mul_le_mul_of_nonneg_right hab.le hT.le) hs] |
| · nlinarith [mul_le_mul_of_nonneg_left |
| (mul_le_mul_of_nonneg_right hab.le hT.le) hr] |
| · ext i |
| fin_cases i < |
| ring |
|
|
| theorem upperIntercept_eq_convexHull {a b T : ℝ} (hab : a < b) (hT : 0 < T) : |
| upperIntercept a b T = convexHull ℝ (Set.range (upperVertices a b T)) := by |
| change upperIntercept a b T = |
| triangle (upperVertices a b T 0) (upperVertices a b T 1) (upperVertices a b T 2) |
| ext z |
| constructor |
| · rintro ⟨x,t,ht0,htT,hax,hxb,rfl⟩ |
| exact upper_triangle_mem hab hT ht0 htT hax hxb |
| · rw [mem_triangle_iff] |
| rintro ⟨r,s,hr,hs,hrs,rfl⟩ |
| refine ⟨r*(a*T)+s*(b*T), r*T+s*T, ?_, ?_, ?_, ?_, ?_⟩ |
| · positivity |
| · nlinarith [mul_le_mul_of_nonneg_right hrs hT.le] |
| · nlinarith [mul_le_mul_of_nonneg_left |
| (mul_le_mul_of_nonneg_right hab.le hT.le) hs] |
| · nlinarith [mul_le_mul_of_nonneg_left |
| (mul_le_mul_of_nonneg_right hab.le hT.le) hr] |
| · ext i |
| fin_cases i < |
| ring |
|
|
| theorem projectedCylinder_carrier_direction (v : Space) (hv : v ≠ 0) |
| (a b c : Space) |
| (h : inner ℝ (unit v) (cross3 (b-a) (c-a)) ≠ 0) : |
| (projectedCylinder (unit v) (unit_norm hv) a b c h).carrier = |
| {z | ∃ p ∈ triangle a b c, ∃ r : ℝ, z = p + r • v} := by |
| ext z |
| constructor |
| · rintro ⟨q,⟨p,hp,rfl⟩,r,hz⟩ |
| refine ⟨p,hp,(r-inner ℝ (unit v) p) / ‖v‖,?_⟩ |
| calc |
| z = projection (unit v) p + r • unit v := hz |
| _ = p + (r-inner ℝ (unit v) p) • unit v := by |
| unfold projection |
| module |
| _ = p + ((r-inner ℝ (unit v) p) / ‖v‖) • v := by |
| simp only [unit, smul_smul, div_eq_mul_inv] |
| · rintro ⟨p,hp,r,hz⟩ |
| exact projectedCylinder_mem_direction v hv a b c h hp r hz |
| |
| theorem lowerCylinder_carrier_intercept {a b T : ℝ} (hab : a < b) (hT : 0 < T) |
| (ε A B : ℝ) : |
| (lowerCylinder hab hT ε A B).carrier = |
| {z | ∃ p ∈ lowerIntercept a b T, ∃ r : ℝ, z = p + r • tiltV ε A B} := by |
| rw [lowerCylinder, projectedCylinder_carrier_direction _ (tiltV_ne_zero ε A B), |
| lowerIntercept_eq_convexHull hab hT] |
| rfl |
|
|
| theorem upperCylinder_carrier_intercept {a b T : ℝ} (hab : a < b) (hT : 0 < T) |
| (ε A B : ℝ) : |
| (upperCylinder hab hT ε A B).carrier = |
| {z | ∃ p ∈ upperIntercept a b T, ∃ r : ℝ, z = p + r • tiltW ε A B} := by |
| rw [upperCylinder, projectedCylinder_carrier_direction _ (tiltW_ne_zero ε A B), |
| upperIntercept_eq_convexHull hab hT] |
| rfl |
| |
| theorem lowerAngularBase_eq_projection_intercept (ε : ℝ) (he : 0 < ε) |
| (j : Fin (n ε)) : |
| lowerAngularBase ε j = projection (lowerAngularAxis ε j) '' |
| lowerIntercept (mesh (n ε) j) (mesh (n ε) (j + 1)) |
| (cutoff ε (mesh (n ε) j) (mesh (n ε) (j + 1))) := by |
| rw [lowerIntercept_eq_convexHull (mesh_strict (N := n ε) (ceil_count_pos he) j) |
| (cutoff_pos_all ε (mesh (n ε) j) (mesh (n ε) (j + 1)))] |
| rfl |
| |
| theorem upperAngularBase_eq_projection_intercept (ε : ℝ) (he : 0 < ε) |
| (j : Fin (n ε)) : |
| upperAngularBase ε j = projection (upperAngularAxis ε j) '' |
| upperIntercept (mesh (n ε) j) (mesh (n ε) (j + 1)) |
| (cutoff ε (mesh (n ε) j) (mesh (n ε) (j + 1))) := by |
| rw [upperIntercept_eq_convexHull (mesh_strict (N := n ε) (ceil_count_pos he) j) |
| (cutoff_pos_all ε (mesh (n ε) j) (mesh (n ε) (j + 1)))] |
| rfl |
| |
| theorem TriCylinder.ext_axis_base {c d : TriCylinder} |
| (ha : c.axis = d.axis) (hb : c.base = d.base) : c = d := by |
| cases c |
| cases d |
| cases ha |
| cases hb |
| rfl |
| |
| theorem finiteFamily_explicit_laws (ε : ℝ) (he : 0 < ε) : |
| ExplicitAngularFamily ε (finiteFamily ε he) := by |
| intro j |
| simp only [lowerAngularIndex, upperAngularIndex, finiteFamily, Function.comp_apply, |
| Fin.cast_cast, Fin.cast_eq_self, Fin.append_left, Fin.append_right] |
| exact ⟨rfl, rfl, rfl, rfl⟩ |
| |
| theorem ExplicitAngularFamily.unique {ε : ℝ} |
| {c d : Fin (2 * n ε) → TriCylinder} |
| (hc : ExplicitAngularFamily ε c) (hd : ExplicitAngularFamily ε d) : c = d := by |
| have h : ∀ k : Fin (n ε + n ε), |
| c (Fin.cast (two_mul (n ε)).symm k) = |
| d (Fin.cast (two_mul (n ε)).symm k) := by |
| intro k |
| refine Fin.addCases (fun j => ?_) (fun j => ?_) k |
| · exact TriCylinder.ext_axis_base ((hc j).1.trans (hd j).1.symm) |
| ((hc j).2.1.trans (hd j).2.1.symm) |
| · exact TriCylinder.ext_axis_base ((hc j).2.2.1.trans (hd j).2.2.1.symm) |
| ((hc j).2.2.2.trans (hd j).2.2.2.symm) |
| funext i |
| simpa only [Fin.cast_cast, Fin.cast_eq_self] using h (Fin.cast (two_mul (n ε)) i) |
| |
| theorem ExplicitAngularFamily.eq_finiteFamily {ε : ℝ} (he : 0 < ε) |
| {c : Fin (2 * n ε) → TriCylinder} (hc : ExplicitAngularFamily ε c) : |
| c = finiteFamily ε he := hc.unique (finiteFamily_explicit_laws ε he) |
| |
| theorem ExplicitAngularFamily.exact_area_sum {ε : ℝ} (he : 0 < ε) |
| {c : Fin (2 * n ε) → TriCylinder} (hc : ExplicitAngularFamily ε c) : |
| totalArea c / H = |
| ∑ j : Fin (n ε), normalizedTerm ε (mesh (n ε) j) (mesh (n ε) (j + 1)) := by |
| rw [hc.eq_finiteFamily he] |
| exact finiteFamily_total ε he |
| |
| theorem ExplicitAngularFamily.covers {ε : ℝ} (he : 0 < ε) |
| (hsmall : ε ≤ 1 / 2000) {c : Fin (2 * n ε) → TriCylinder} |
| (hc : ExplicitAngularFamily ε c) : Covers c := by |
| rw [hc.eq_finiteFamily he] |
| apply finiteFamily_covers he |
| norm_num [η] at * |
| exact hsmall |
| |
| end TriangularCovering |
| |
| end |
| |
| end OAI |
| |
