| import OAI.Geometry.CylinderCovering.Barycentric.CanonicalMapDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.PairedActual |
|
|
| namespace OAI |
|
|
| open Set |
| open scoped InnerProductSpace |
| noncomputable section |
| namespace RadialCoordinates |
| open CylinderCovering |
|
|
| private lemma Hpos : 0 < Real.sqrt (2:ℝ) := Real.sqrt_pos.mpr (by norm_num) |
| private lemma Hne : Real.sqrt (2:ℝ) ≠ 0 := Hpos.ne' |
| private lemma Hsq : (Real.sqrt (2:ℝ))^2 = 2 := Real.sq_sqrt (by norm_num) |
| |
| theorem canonicalEmbedding_norm (z : Space) : ‖canonicalEmbedding z‖ = ‖z‖ := by |
| have hs : ‖canonicalEmbedding z‖^2 = ‖z‖^2 := by |
| simp only [EuclideanSpace.real_norm_sq_eq,Fin.sum_univ_four,Fin.sum_univ_three] |
| simp only [canonicalEmbedding,LinearMap.coe_mk,AddHom.coe_mk] |
| dsimp |
| field_simp |
| nlinarith [Hsq] |
| nlinarith [norm_nonneg (canonicalEmbedding z),norm_nonneg z] |
| |
| def canonicalEmbeddingIsometry : Space →ₗᵢ[ℝ] BarycentricSpace := |
| {canonicalEmbedding with norm_map' := canonicalEmbedding_norm} |
|
|
| theorem canonicalScale_pos : 0 < canonicalScale := by |
| unfold canonicalScale |
| positivity |
|
|
| theorem canonicalToBary_eq (z : Space) : |
| canonicalToBary z = canonicalCenter + canonicalScale • canonicalEmbedding z := by |
| ext i |
| fin_cases i <;> |
| simp [canonicalToBary,barycentricPoint,canonicalCenter,canonicalScale,canonicalEmbedding] <;> |
| field_simp <;> ring_nf <;> simp <;> ring |
|
|
| theorem angular_point (N j : ℕ) : |
| angular (RadialRiemann.point 0 1 N j) = RadialRiemann.point (-1) 1 N j := by |
| unfold angular RadialRiemann.point RadialRiemann.step |
| ring |
|
|
| theorem angular_mem_cell_iff (N : ℕ) (i : Fin N) (t : ℝ) : |
| angular t ∈ RadialRiemann.cell (-1) 1 N i ↔ t ∈ RadialRiemann.cell 0 1 N i := by |
| unfold RadialRiemann.cell |
| rw [← angular_point N i,← angular_point N (i+1)] |
| simp only [mem_Icc,angular] |
| constructor <;> rintro ⟨h1,h2⟩ <;> constructor <;> linarith |
|
|
| theorem angular_tags (tag : Fin N → ℝ) |
| (htag : RadialRiemann.IsTagged 0 1 N tag) : |
| RadialRiemann.IsTagged (-1) 1 N (fun i => angular (tag i)) := |
| fun i => (angular_mem_cell_iff N i (tag i)).mpr (htag i) |
|
|
| theorem angular_inverse (q : ℝ) : angular ((q+1)/2) = q := by unfold angular; ring |
|
|
| theorem inverse_angular (t : ℝ) : (angular t+1)/2 = t := by unfold angular; ring |
|
|
| theorem lowerPoint_toBary (k t S p : ℝ) : |
| canonicalToBary (RadialPairedSweep.point |
| ((2*S)*angular t+k*(2*p)*RadialCaps.G (angular t)) (2*p) |
| (Real.sqrt 2*(-1+2*S+k*(2*p)*angular t))) = firstSweepPoint k t S p := by |
| ext i |
| fin_cases i <;> |
| simp [canonicalToBary,RadialPairedSweep.point,firstSweepPoint, |
| barycentricPoint,RadialCaps.G] <;> field_simp [Hne] <;> ring |
|
|
| theorem upperPoint_toBary (k t S p : ℝ) : |
| canonicalToBary (RadialPairedSweep.point (2*p) |
| ((2*S)*angular t-k*(2*p)*RadialCaps.G (angular t)) |
| (Real.sqrt 2*(1-2*S+k*(2*p)*angular t))) = secondSweepPoint k t S p := by |
| ext i |
| fin_cases i <;> |
| simp [canonicalToBary,RadialPairedSweep.point,secondSweepPoint, |
| barycentricPoint,RadialCaps.G] <;> field_simp [Hne] <;> ring |
|
|
| theorem lowerLocalSweep_image (k : ℝ) (N : ℕ) (i : Fin N) : |
| canonicalToBary '' RadialPairedSweep.lowerLocalSweep (Real.sqrt 2) k |
| (canonicalCap k) N i = BarycentricFinite.firstLocalSweep k N i := by |
| ext z |
| constructor |
| · rintro ⟨_,⟨q,hq,ρ,hρ,y,hy,rfl⟩,rfl⟩ |
| have ht : (q+1)/2 ∈ RadialRiemann.cell 0 1 N i := by |
| apply (angular_mem_cell_iff N i _).mp |
| simpa only [angular_inverse] using hq |
| refine ⟨(q+1)/2,ht,ρ/2,⟨by linarith [hρ.1],?_⟩, |
| y/2,⟨by linarith [(abs_le.mp hy).1],by linarith [(abs_le.mp hy).2]⟩,?_⟩ |
| · change ρ/2 ≤ cap k ((q+1)/2) |
| have hc := hρ.2 |
| change ρ ≤ 2*cap k ((q+1)/2) at hc |
| linarith |
| · have hr : 2*(ρ/2)=ρ := by ring |
| have hy : 2*(y/2)=y := by ring |
| simpa only [angular_inverse,hr,hy] using lowerPoint_toBary k ((q+1)/2) (ρ/2) (y/2) |
| · rintro ⟨t,ht,S,hS,p,hp,rfl⟩ |
| refine ⟨RadialPairedSweep.point |
| ((2*S)*angular t+k*(2*p)*RadialCaps.G (angular t)) (2*p) |
| (Real.sqrt 2*(-1+2*S+k*(2*p)*angular t)),?_,lowerPoint_toBary k t S p⟩ |
| refine ⟨angular t,(angular_mem_cell_iff N i t).mpr ht,2*S, |
| ⟨by linarith [hS.1],?_⟩,2*p,?_,rfl⟩ |
| · simp only [canonicalCap,inverse_angular] |
| linarith [hS.2] |
| · rw [abs_le] |
| constructor <;> linarith [hp.1,hp.2] |
|
|
| theorem upperLocalSweep_image (k : ℝ) (N : ℕ) (i : Fin N) : |
| canonicalToBary '' RadialPairedSweep.upperLocalSweep (Real.sqrt 2) k |
| (canonicalCap k) N i = BarycentricFinite.secondLocalSweep k N i := by |
| ext z |
| constructor |
| · rintro ⟨_,⟨q,hq,ρ,hρ,y,hy,rfl⟩,rfl⟩ |
| have ht : (q+1)/2 ∈ RadialRiemann.cell 0 1 N i := by |
| apply (angular_mem_cell_iff N i _).mp |
| simpa only [angular_inverse] using hq |
| refine ⟨(q+1)/2,ht,ρ/2,⟨by linarith [hρ.1],?_⟩, |
| y/2,⟨by linarith [(abs_le.mp hy).1],by linarith [(abs_le.mp hy).2]⟩,?_⟩ |
| · change ρ/2 ≤ cap k ((q+1)/2) |
| have hc := hρ.2 |
| change ρ ≤ 2*cap k ((q+1)/2) at hc |
| linarith |
| · have hr : 2*(ρ/2)=ρ := by ring |
| have hy : 2*(y/2)=y := by ring |
| simpa only [angular_inverse,hr,hy] using upperPoint_toBary k ((q+1)/2) (ρ/2) (y/2) |
| · rintro ⟨t,ht,S,hS,p,hp,rfl⟩ |
| refine ⟨RadialPairedSweep.point (2*p) |
| ((2*S)*angular t-k*(2*p)*RadialCaps.G (angular t)) |
| (Real.sqrt 2*(1-2*S+k*(2*p)*angular t)),?_,upperPoint_toBary k t S p⟩ |
| refine ⟨angular t,(angular_mem_cell_iff N i t).mpr ht,2*S, |
| ⟨by linarith [hS.1],?_⟩,2*p,?_,rfl⟩ |
| · simp only [canonicalCap,inverse_angular] |
| linarith [hS.2] |
| · rw [abs_le] |
| constructor <;> linarith [hp.1,hp.2] |
|
|
| theorem canonicalEmbedding_lower_raw (k t : ℝ) : |
| canonicalEmbedding (RadialPairedSweep.point (k*RadialCaps.G (angular t)) 1 |
| (Real.sqrt 2*k*angular t)) = Real.sqrt 2 • firstDirection k t := by |
| ext i |
| fin_cases i <;> |
| simp [canonicalEmbedding,RadialPairedSweep.point,firstDirection,RadialCaps.G,angular] <;> |
| field_simp [Hne] <;> ring_nf <;> simp <;> ring |
|
|
| theorem canonicalEmbedding_upper_raw (k t : ℝ) : |
| canonicalEmbedding (RadialPairedSweep.point 1 (-k*RadialCaps.G (angular t)) |
| (Real.sqrt 2*k*angular t)) = Real.sqrt 2 • BarycentricFinite.secondDirection k t := by |
| ext i |
| fin_cases i <;> |
| simp [canonicalEmbedding,RadialPairedSweep.point,BarycentricFinite.secondDirection, |
| RadialCaps.G,angular] <;> |
| field_simp [Hne] <;> ring_nf <;> simp <;> ring |
|
|
| private theorem normalized_positive_smul (v : BarycentricSpace) : |
| ‖Real.sqrt 2 • v‖⁻¹ • (Real.sqrt 2 • v) = ‖v‖⁻¹ • v := by |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos Hpos,smul_smul] |
| congr 1 |
| by_cases hn : ‖v‖ = 0 |
| · simp [hn] |
| · field_simp [Hne,hn] |
|
|
| theorem canonicalEmbedding_lowerAxis (k t : ℝ) : |
| canonicalEmbedding (RadialPairedSweep.lowerAxis (Real.sqrt 2) k (angular t)) = |
| BarycentricFinite.firstAxis k t := by |
| unfold RadialPairedSweep.lowerAxis |
| dsimp only |
| rw [map_smul,← canonicalEmbedding_norm,canonicalEmbedding_lower_raw] |
| exact normalized_positive_smul _ |
|
|
| theorem canonicalEmbedding_upperAxis (k t : ℝ) : |
| canonicalEmbedding (RadialPairedSweep.upperAxis (Real.sqrt 2) k (angular t)) = |
| BarycentricFinite.secondAxis k t := by |
| unfold RadialPairedSweep.upperAxis |
| dsimp only |
| rw [map_smul,← canonicalEmbedding_norm,canonicalEmbedding_upper_raw] |
| exact normalized_positive_smul _ |
|
|
| theorem canonicalScale_sq : canonicalScale^2 = (1/8 : ℝ) := by |
| unfold canonicalScale |
| rw [div_pow,mul_pow,Hsq] |
| norm_num |
|
|
| theorem canonicalCap_continuous (k : ℝ) : Continuous (canonicalCap k) := by |
| unfold canonicalCap cap capPolynomial angular |
| fun_prop |
|
|
| theorem canonicalCap_nonneg (k q : ℝ) : 0 ≤ canonicalCap k q := by |
| unfold canonicalCap cap capPolynomial |
| positivity |
|
|
| theorem barycentric_midpoint_tagged (N : ℕ) : |
| RadialRiemann.IsTagged 0 1 N (BarycentricFinite.midpoint N) := by |
| intro i |
| obtain ⟨t,ht⟩ := RadialRiemann.cell_nonempty (show (0:ℝ)<1 by norm_num) i |
| have hle := ht.1.trans ht.2 |
| change RadialRiemann.point 0 1 N i ≤ _ ∧ _ ≤ RadialRiemann.point 0 1 N (i+1) |
| unfold BarycentricFinite.midpoint |
| constructor <;> linarith |
|
|
| theorem canonical_midpoint_tagged (N : ℕ) : |
| RadialRiemann.IsTagged (-1) 1 N (fun i => angular (BarycentricFinite.midpoint N i)) := |
| angular_tags _ (barycentric_midpoint_tagged N) |
|
|
| theorem canonicalCap_eq (k q : ℝ) : |
| canonicalCap k q = 1+k^2*((q^2+q^4)/2+1/500) := by |
| unfold canonicalCap cap capPolynomial angular |
| ring |
|
|
| theorem canonical_physicalCost (k : ℝ) : |
| RadialPairedSweep.physicalCost (Real.sqrt 2) k (canonicalCap k) = |
| ∫ q in (-1:ℝ)..1, |
| (1+k^2*((q^2+q^4)/2+1/500))^2 / |
| (2*Real.sqrt (1+k^2*(2*q^2+((1+q^2)/2)^2))) := by |
| unfold RadialPairedSweep.physicalCost |
| apply intervalIntegral.integral_congr |
| intro q _ |
| dsimp only |
| rw [canonicalCap_eq] |
| have hd : RadialCaps.G q^2+(Real.sqrt 2)^2*q^2 = |
| 2*q^2+((1+q^2)/2)^2 := by rw [Hsq]; unfold RadialCaps.G; ring |
| rw [hd] |
|
|
| end RadialCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
