| import OAI.Combinatorics.CompleteCrossing.Polygonalization |
| import OAI.Combinatorics.Crossing.CrossingReconnection |
|
|
| namespace OAI |
|
|
| /-! Crossing reconnection chords and avoidance of vertices and retained arcs. -/ |
|
|
| section |
|
|
| /-! The two actual disjoint chords used in an adjacent-crossing surgery disk. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.DrawingPassages |
| variable {D : Paper170.AdmissibleDrawing graph} |
|
|
| def reconnectionStart (C : DrawingPassages D.toFiniteIntersectionDrawing) (p q : C.InnerIndex) (i : Bool) : Plane := |
| if i then C.portPoint (C.incomingPort p) else C.portPoint (C.incomingPort q) |
|
|
| def reconnectionEnd (C : DrawingPassages D.toFiniteIntersectionDrawing) (p q : C.InnerIndex) (i : Bool) : Plane := |
| if i then C.portPoint (C.outgoingPort q) else C.portPoint (C.outgoingPort p) |
|
|
| theorem reconnection_endpoints_ne (C : DrawingPassages D.toFiniteIntersectionDrawing) (p q : C.InnerIndex) (i : Bool) : |
| C.reconnectionStart p q i ≠ C.reconnectionEnd p q i := by |
| cases i |
| · exact C.portPoint_injective.ne (C.incomingPort_ne_outgoingPort q p) |
| · exact C.portPoint_injective.ne (C.incomingPort_ne_outgoingPort p q) |
|
|
| theorem crossingChord_meeting_segmentsCrossProperly (C : DrawingPassages D.toFiniteIntersectionDrawing) (p q : C.InnerIndex) |
| (hpq : p ≠ q) (x : Plane) (hxp : x ∈ Set.range (C.crossingChord p)) |
| (hxq : x ∈ Set.range (C.crossingChord q)) : |
| SegmentsCrossProperly (C.portPoint (C.incomingPort p)) (C.portPoint (C.outgoingPort p)) |
| (C.portPoint (C.incomingPort q)) (C.portPoint (C.outgoingPort q)) := by |
| have hm := C.chord_meeting_marks_eq p q hxp hxq |
| obtain ⟨s,hs⟩ := hxp |
| obtain ⟨t,ht⟩ := hxq |
| have he : segmentPoint (C.portPoint (C.incomingPort p)) (C.portPoint (C.outgoingPort p)) s.val= |
| segmentPoint (C.portPoint (C.incomingPort q)) (C.portPoint (C.outgoingPort q)) t.val := by |
| simpa only [crossingChord,Path.segment_apply,AffineMap.lineMap_apply_module,segmentPoint] using hs.trans ht.symm |
| obtain ⟨hd,hsi,hti⟩ := C.crossing_chords_transverse_at_mark p q hpq hm s.val s.property t.val t.property he |
| exact ⟨hd,s.val,hsi,t.val,hti,he⟩ |
|
|
| theorem reconnection_chords_disjoint (C : DrawingPassages D.toFiniteIntersectionDrawing) (p q : C.InnerIndex) |
| (hcross : SegmentsCrossProperly (C.portPoint (C.incomingPort p)) (C.portPoint (C.outgoingPort p)) |
| (C.portPoint (C.incomingPort q)) (C.portPoint (C.outgoingPort q))) : |
| ∀ i j : Bool, i ≠ j → Disjoint |
| (segment ℝ (C.reconnectionStart p q i) (C.reconnectionEnd p q i)) |
| (segment ℝ (C.reconnectionStart p q j) (C.reconnectionEnd p q j)) := by |
| have hh := crossing_reconnections_disjoint _ _ _ _ hcross |
| intro i j hij |
| cases i <;> cases j |
| · exact False.elim (hij rfl) |
| · exact hh |
| · exact hh.symm |
| · exact False.elim (hij rfl) |
| |
| theorem reconnection_endpoints_on_circle (C : DrawingPassages D.toFiniteIntersectionDrawing) (p q : C.InnerIndex) |
| (hm : C.innerMark p=C.innerMark q) (i : Bool) : |
| squaredLength (C.reconnectionStart p q i-(C.innerMark p).val)=C.radius^2 ∧ |
| squaredLength (C.reconnectionEnd p q i-(C.innerMark p).val)=C.radius^2 := by |
| cases i |
| · change squaredLength (C.portPoint (C.incomingPort q)-(C.innerMark p).val)=C.radius^2 ∧ |
| squaredLength (C.portPoint (C.outgoingPort p)-(C.innerMark p).val)=C.radius^2 |
| exact ⟨by simpa only [hm] using C.incomingPoint_on_circle q,C.outgoingPoint_on_circle p⟩ |
| · change squaredLength (C.portPoint (C.incomingPort p)-(C.innerMark p).val)=C.radius^2 ∧ |
| squaredLength (C.portPoint (C.outgoingPort q)-(C.innerMark p).val)=C.radius^2 |
| exact ⟨C.incomingPoint_on_circle p,by simpa only [hm] using C.outgoingPoint_on_circle q⟩ |
|
|
| theorem reconnection_open_in_disk (C : DrawingPassages D.toFiniteIntersectionDrawing) (p q : C.InnerIndex) |
| (hm : C.innerMark p=C.innerMark q) (i : Bool) : |
| openSegmentTrace (C.reconnectionStart p q i) (C.reconnectionEnd p q i) ⊆ C.crossingDisk p := by |
| rintro x ⟨t,ht,rfl⟩ |
| exact round_chord_in_open_disk _ _ _ _ (C.reconnection_endpoints_on_circle p q hm i).1 |
| (C.reconnection_endpoints_on_circle p q hm i).2 (C.reconnection_endpoints_ne p q i) t ht |
|
|
| end Paper170Geometry.DrawingPassages |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! The new closed chords stay in their crossing disk and avoid all placed vertices. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.DrawingPassages |
| variable {D : Paper170.AdmissibleDrawing graph} |
|
|
| theorem reconnection_segment_in_disk (C : DrawingPassages D.toFiniteIntersectionDrawing) (p q : C.InnerIndex) |
| (hm : C.innerMark p=C.innerMark q) (i : Bool) : |
| segment ℝ (C.reconnectionStart p q i) (C.reconnectionEnd p q i) ⊆ |
| closedRoundDisk (C.innerMark p).val C.radius := by |
| intro x hx |
| rw [← Path.range_segment] at hx |
| obtain ⟨t,ht⟩ := hx |
| rw [← ht] |
| change squaredLength ((Path.segment _ _) t-(C.innerMark p).val) ≤ C.radius^2 |
| simp only [Path.segment_apply,AffineMap.lineMap_apply_module] |
| exact round_chord_in_disk _ _ _ _ (C.reconnection_endpoints_on_circle p q hm i).1 |
| (C.reconnection_endpoints_on_circle p q hm i).2 t.val t.property |
|
|
| theorem reconnection_segment_avoids_vertices (C : DrawingPassages D.toFiniteIntersectionDrawing) (p q : C.InnerIndex) |
| (hm : C.innerMark p=C.innerMark q) (v : Vertex → Plane) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) (i : Bool) (j : Vertex) : |
| v j ∉ segment ℝ (C.reconnectionStart p q i) (C.reconnectionEnd p q i) := by |
| intro hx |
| have hn : C.innerMark p ≠ C.vertexMark j := by |
| intro he |
| exact crossing_not_vertex D.toFiniteIntersectionDrawing (C.innerMark p).val (C.innerMark_is_crossing p) j (congrArg Subtype.val he) |
| exact Set.disjoint_left.mp (C.disks_disjoint _ _ hn) |
| (C.reconnection_segment_in_disk p q hm i hx) (hv j).le |
|
|
| end Paper170Geometry.DrawingPassages |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Neither reconnection segment contains the actual point being removed. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
|
|
| theorem segment_avoids_of_affine_positive (f : Plane → ℝ) (a b x : Plane) |
| (hf : ∀ t, f (segmentPoint a b t)=(1-t)*f a+t*f b) |
| (ha : 0 < f a) (hb : 0 < f b) (hx : f x=0) : x ∉ segment ℝ a b := by |
| intro h |
| rw [← Path.range_segment] at h |
| obtain ⟨t,ht⟩ := h |
| have ht' : segmentPoint a b t.val=x := by |
| simpa only [Path.segment_apply,AffineMap.lineMap_apply_module,segmentPoint] using ht |
| have hp := convex_combination_pos (f a) (f b) t.val ha hb t.property |
| rw [← hf t.val,ht',hx] at hp |
| exact lt_irrefl 0 hp |
|
|
| theorem crossing_reconnections_avoid_point (a b c d x : Plane) |
| (hd : det2 (b-a) (d-c) ≠ 0) (s t : ℝ) |
| (hs : s ∈ Set.Ioo (0 : ℝ) 1) (ht : t ∈ Set.Ioo (0 : ℝ) 1) |
| (hx : x=segmentPoint a b s) (hy : x=segmentPoint c d t) : |
| x ∉ segment ℝ c b ∧ x ∉ segment ℝ a d := by |
| let F := crossingCoordinates x (b-a) (d-c) |
| have hFa : F a=(-s,0) := by |
| simpa only [segmentPoint_zero,zero_sub] using crossingCoordinates_first_segment a b c d x s 0 hd hx |
| have hFb : F b=(1-s,0) := by |
| simpa only [segmentPoint_one] using crossingCoordinates_first_segment a b c d x s 1 hd hx |
| have hFc : F c=(0,-t) := by |
| simpa only [segmentPoint_zero,zero_sub] using crossingCoordinates_second_segment a b c d x t 0 hd hy |
| have hFd : F d=(0,1-t) := by |
| simpa only [segmentPoint_one] using crossingCoordinates_second_segment a b c d x t 1 hd hy |
| have hF (u v : Plane) (r : ℝ) : F (segmentPoint u v r)=segmentPoint (F u) (F v) r := |
| crossingCoordinates_affine x (b-a) (d-c) u v r |
| have hzero : F x=(0,0) := crossingCoordinates_origin x (b-a) (d-c) |
| constructor |
| · apply segment_avoids_of_affine_positive (fun z => (F z).1-(F z).2) c b x |
| · intro r |
| rw [hF] |
| dsimp [segmentPoint] |
| ring |
| · rw [hFc] |
| dsimp |
| linarith [ht.1] |
| · rw [hFb] |
| dsimp |
| linarith [hs.2] |
| · rw [hzero] |
| norm_num |
| · apply segment_avoids_of_affine_positive (fun z => (F z).2-(F z).1) a d x |
| · intro r |
| rw [hF] |
| dsimp [segmentPoint] |
| ring |
| · rw [hFa] |
| dsimp |
| linarith [hs.1] |
| · rw [hFd] |
| dsimp |
| linarith [ht.2] |
| · rw [hzero] |
| norm_num |
|
|
| theorem DrawingPassages.reconnection_avoids_chord_meeting {D : Paper170.AdmissibleDrawing graph} |
| (C : DrawingPassages D.toFiniteIntersectionDrawing) (p q : C.InnerIndex) (hpq : p ≠ q) (x : Plane) |
| (hxp : x ∈ Set.range (C.crossingChord p)) (hxq : x ∈ Set.range (C.crossingChord q)) : |
| x ∈ C.crossingDisk p ∧ ∀ i : Bool, |
| x ∉ segment ℝ (C.reconnectionStart p q i) (C.reconnectionEnd p q i) := by |
| have hm := C.chord_meeting_marks_eq p q hxp hxq |
| obtain ⟨s,hs⟩ := hxp |
| obtain ⟨t,ht⟩ := hxq |
| have hs' : segmentPoint (C.portPoint (C.incomingPort p)) (C.portPoint (C.outgoingPort p)) s.val=x := by |
| simpa only [DrawingPassages.crossingChord,Path.segment_apply,AffineMap.lineMap_apply_module,segmentPoint] using hs |
| have ht' : segmentPoint (C.portPoint (C.incomingPort q)) (C.portPoint (C.outgoingPort q)) t.val=x := by |
| simpa only [DrawingPassages.crossingChord,Path.segment_apply,AffineMap.lineMap_apply_module,segmentPoint] using ht |
| obtain ⟨hd,hsi,hti⟩ := C.crossing_chords_transverse_at_mark p q hpq hm s.val s.property t.val t.property |
| (hs'.trans ht'.symm) |
| constructor |
| · change squaredLength (x-(C.innerMark p).val) < C.radius^2 |
| rw [← hs] |
| exact C.crossingChord_interior_in_disk p s hsi |
| · have ha := crossing_reconnections_avoid_point _ _ _ _ x hd s.val t.val hsi hti hs'.symm ht'.symm |
| intro i |
| cases i |
| · exact ha.1 |
| · exact ha.2 |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| end OAI |
|
|
