| import OAI.Geometry.CylinderCovering.Barycentric.Sector |
| import OAI.Geometry.CylinderCovering.Beams.Neighborhood |
| import OAI.Geometry.CylinderCovering.Beams.FullDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.Riemann |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory |
| namespace CylinderCovering.BeamNeighborhood |
|
|
| lemma sector_eq_signedSector (a l r δ : ℝ) : |
| sector a l r δ = |
| RadialCoordinates.BarycentricSector.signedSector l r δ (2-a+δ) := rfl |
|
|
| theorem sector_eq_two_triangles {a l r δ : ℝ} |
| (ha : a ≤ 2) (hlr : l ≤ r) (hδ : 0 ≤ δ) : |
| sector a l r δ = |
| convexHull ℝ (range (RadialCoordinates.BarycentricSector.vertices l r (2-a+δ))) ∪ |
| (fun z : Plane => -z) '' |
| convexHull ℝ (range (RadialCoordinates.BarycentricSector.vertices l r δ)) := by |
| rw [sector_eq_signedSector] |
| exact RadialCoordinates.BarycentricSector.signedSector_eq_two_triangles |
| hlr hδ (by linarith) |
|
|
| theorem sector_area {a l r δ : ℝ} |
| (ha : a ≤ 2) (hlr : l ≤ r) (hδ : 0 ≤ δ) : |
| volume.real (sector a l r δ) = ((2-a+δ)^2+δ^2)/2*(r-l) := by |
| rw [sector_eq_signedSector, |
| RadialCoordinates.BarycentricSector.signedSector_area hlr hδ (by linarith)] |
| ring |
|
|
| theorem sector_perimeters_le {a l r δ : ℝ} |
| (ha : 0 ≤ a) (ha' : a ≤ 2) (hl : 0 ≤ l) (hlr : l ≤ r) (hr : r ≤ 1) |
| (hδ : 0 ≤ δ) (hδ' : δ ≤ 1) : |
| RadialCoordinates.BarycentricSector.perimeter l r δ + |
| RadialCoordinates.BarycentricSector.perimeter l r (2-a+δ) ≤ |
| 4*(2+Real.sqrt 2) := by |
| have h₁ := RadialCoordinates.BarycentricSector.perimeter_le hl hlr hr hδ |
| have h₂ := RadialCoordinates.BarycentricSector.perimeter_le hl hlr hr |
| (show 0 ≤ 2-a+δ by linarith) |
| have hs := Real.sqrt_nonneg (2:ℝ) |
| have hm := mul_le_mul_of_nonneg_left (show 2-a+2*δ ≤ 4 by linarith) |
| (show 0 ≤ 2+Real.sqrt 2 by positivity) |
| nlinarith |
|
|
| theorem sector_area_error {a l r δ : ℝ} |
| (ha : 0 ≤ a) (ha' : a ≤ 2) (hδ : 0 ≤ δ) (hδ' : δ ≤ 1) |
| (hlen : r-l = δ) : |
| 0 ≤ volume.real (sector a l r δ) - (2-a)^2/2*δ ∧ |
| volume.real (sector a l r δ) - (2-a)^2/2*δ ≤ 3*δ^2 := by |
| rw [sector_area ha' (by linarith) hδ, hlen] |
| have hδ3 : δ^3 ≤ δ^2 := by |
| nlinarith [mul_le_mul_of_nonneg_right hδ' (sq_nonneg δ)] |
| have hnonneg : 0 ≤ (2-a)*δ^2+δ^3 := by positivity |
| have hbound : (2-a)*δ^2 ≤ 2*δ^2 := by nlinarith [sq_nonneg δ] |
| constructor <;> nlinarith |
|
|
| end CylinderCovering.BeamNeighborhood |
|
|
| namespace ShortenedBeam |
| open CylinderCovering |
|
|
| lemma doubleSector_eq_signedSectorModel (a : ℝ) (N : ℕ) (i : Fin N) : |
| doubleSector a N i = BeamNeighborhood.sector a |
| (RadialRiemann.point 0 1 N i) (RadialRiemann.point 0 1 N (i+1)) (1/(N:ℝ)) := by |
| have hv (d p : ℝ) : (!₂[d*p,d*(1-p)] : ParameterPlane) = |
| d • (!₂[p,1-p] : ParameterPlane) := by |
| ext j |
| fin_cases j <;> rfl |
| ext z |
| simp only [doubleSector, BeamNeighborhood.sector, RadialRiemann.cell, |
| BeamNeighborhood.radial, hv] |
|
|
| theorem doubleSector_area {a : ℝ} (ha : a ≤ 2) |
| (N : ℕ) (i : Fin N) : |
| volume.real (doubleSector a N i) = |
| ((2-a+1/(N:ℝ))^2+(1/(N:ℝ))^2)/2*(1/(N:ℝ)) := by |
| have hδ : 0 ≤ 1/(N:ℝ) := by positivity |
| have hlen : RadialRiemann.point 0 1 N (i+1) - RadialRiemann.point 0 1 N i = |
| 1/(N:ℝ) := by |
| rw [RadialRiemann.point_succ] |
| simp [RadialRiemann.step] |
| rw [doubleSector_eq_signedSectorModel, BeamNeighborhood.sector_area ha (by linarith) hδ, hlen] |
|
|
| end ShortenedBeam |
|
|
| end |
|
|
| end OAI |
|
|
