| import OAI.Geometry.CylinderCovering.Caps.FiniteDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.SetProjection |
| import OAI.Geometry.CylinderCovering.Footprints.Area |
| import OAI.Geometry.CylinderCovering.Radial.Riemann |
|
|
| namespace OAI |
|
|
| open Set MeasureTheory |
| open scoped InnerProductSpace |
| noncomputable section |
| namespace CylinderCovering.HalfheightFinite |
| open TriangularCovering |
|
|
| private def normalFirst : Space := !₂[1, 0, 0] |
| private def normalSecond : Space := !₂[0, 1, 0] |
| private def radialFirst : Space := !₂[0, 0, height] |
| private def radialSecond : Space := !₂[0, 0, -height] |
| private def transverseFirst : Space := !₂[0, 1/2, 0] |
| private def transverseSecond : Space := !₂[1/2, 0, 0] |
| private def originSecond : Space := !₂[0, 0, height] |
| private def velocityFirst (δ r : ℝ) : Space := !₂[0, δ*alpha r, 2*height*δ*beta r] |
| private def velocitySecond (δ r : ℝ) : Space := !₂[-δ*alpha r, 0, 2*height*δ*beta r] |
|
|
| private lemma height_positive : 0 < height := by unfold height; positivity |
| private lemma height_squared : height ^ 2 = 1/2 := by norm_num [height, div_pow] |
|
|
| private lemma normalFirst_unit : ‖normalFirst‖ = 1 := by |
| rw [← Real.sqrt_sq (norm_nonneg _), EuclideanSpace.real_norm_sq_eq] |
| norm_num [normalFirst, Fin.sum_univ_succ] |
| private lemma normalSecond_unit : ‖normalSecond‖ = 1 := by |
| rw [← Real.sqrt_sq (norm_nonneg _), EuclideanSpace.real_norm_sq_eq] |
| norm_num [normalSecond, Fin.sum_univ_succ] |
|
|
| private lemma first_chart : |
| (RadialProjection.coordinateAffine 0 radialFirst transverseFirst : (ℝ × ℝ) → Space) = firstFoot := by |
| funext p |
| ext i |
| fin_cases i <;> simp [RadialProjection.coordinateAffine, radialFirst, transverseFirst, firstFoot] <;> ring |
| private lemma second_chart : |
| (RadialProjection.coordinateAffine originSecond radialSecond transverseSecond : (ℝ × ℝ) → Space) = secondFoot := by |
| funext p |
| ext i |
| fin_cases i <;> simp [RadialProjection.coordinateAffine, originSecond, radialSecond, transverseSecond, secondFoot] <;> ring |
|
|
| private lemma first_raw (δ r : ℝ) : normalFirst + velocityFirst δ r = firstRaw δ r := by |
| ext i |
| fin_cases i <;> simp [normalFirst, velocityFirst, firstRaw] |
| private lemma second_raw (δ r : ℝ) : normalSecond + velocitySecond δ r = secondRaw δ r := by |
| ext i |
| fin_cases i <;> simp [normalSecond, velocitySecond, secondRaw] |
|
|
| private lemma first_gram : |
| Real.sqrt (inner ℝ radialFirst radialFirst * inner ℝ transverseFirst transverseFirst - |
| (inner ℝ radialFirst transverseFirst)^2) = height/2 := by |
| have h : inner ℝ radialFirst radialFirst * inner ℝ transverseFirst transverseFirst - |
| (inner ℝ radialFirst transverseFirst)^2 = (height/2)^2 := by |
| simp only [inner_coord, radialFirst, transverseFirst] |
| dsimp |
| ring |
| rw [h, Real.sqrt_sq (div_nonneg height_positive.le (by norm_num))] |
| private lemma second_gram : |
| Real.sqrt (inner ℝ radialSecond radialSecond * inner ℝ transverseSecond transverseSecond - |
| (inner ℝ radialSecond transverseSecond)^2) = height/2 := by |
| have h : inner ℝ radialSecond radialSecond * inner ℝ transverseSecond transverseSecond - |
| (inner ℝ radialSecond transverseSecond)^2 = (height/2)^2 := by |
| simp only [inner_coord, radialSecond, transverseSecond] |
| dsimp |
| ring |
| rw [h, Real.sqrt_sq (div_nonneg height_positive.le (by norm_num))] |
|
|
| private lemma velocityFirst_norm_sq (δ r : ℝ) : |
| ‖velocityFirst δ r‖^2 = δ^2*(alpha r^2+2*beta r^2) := by |
| rw [EuclideanSpace.real_norm_sq_eq] |
| simp [velocityFirst, Fin.sum_univ_succ] |
| nlinarith only [height_squared] |
| private lemma velocitySecond_norm_sq (δ r : ℝ) : |
| ‖velocitySecond δ r‖^2 = δ^2*(alpha r^2+2*beta r^2) := by |
| rw [EuclideanSpace.real_norm_sq_eq] |
| simp [velocitySecond, Fin.sum_univ_succ] |
| nlinarith only [height_squared] |
|
|
| theorem firstFoot_projection_area (δ r : ℝ) (S : Set (ℝ × ℝ)) : |
| area (projection (firstAxis δ r) '' (firstFoot '' S)) = |
| (height/2) / Real.sqrt (1+δ^2*(alpha r^2+2*beta r^2)) * volume.real S := by |
| have h := RadialProjection.radial_affine_projection_area 0 normalFirst radialFirst transverseFirst |
| (velocityFirst δ r) normalFirst_unit |
| (by simp [inner_coord, normalFirst, radialFirst]) |
| (by simp [inner_coord, normalFirst, transverseFirst]) |
| (by simp [inner_coord, normalFirst, velocityFirst]) S |
| rw [first_chart, first_raw, first_gram, velocityFirst_norm_sq] at h |
| exact h |
|
|
| theorem secondFoot_projection_area (δ r : ℝ) (S : Set (ℝ × ℝ)) : |
| area (projection (secondAxis δ r) '' (secondFoot '' S)) = |
| (height/2) / Real.sqrt (1+δ^2*(alpha r^2+2*beta r^2)) * volume.real S := by |
| have h := RadialProjection.radial_affine_projection_area originSecond normalSecond radialSecond transverseSecond |
| (velocitySecond δ r) normalSecond_unit |
| (by simp [inner_coord, normalSecond, radialSecond]) |
| (by simp [inner_coord, normalSecond, transverseSecond]) |
| (by simp [inner_coord, normalSecond, velocitySecond]) S |
| rw [second_chart, second_raw, second_gram, velocitySecond_norm_sq] at h |
| exact h |
|
|
| theorem firstFoot_projection_finite (δ r : ℝ) {S : Set (ℝ × ℝ)} (hS : IsCompact S) : |
| Measure.euclideanHausdorffMeasure 2 |
| (projection (firstAxis δ r) '' (firstFoot '' S)) ≠ ⊤ := by |
| have hp : inner ℝ normalFirst (velocityFirst δ r) = 0 := by |
| simp [inner_coord, normalFirst, velocityFirst] |
| have h := RadialProjection.affine_projection_finite 0 |
| (unit (normalFirst + velocityFirst δ r)) radialFirst transverseFirst |
| (unit_norm (RadialProjection.sideways_ne_zero normalFirst_unit hp)) hS |
| rw [first_chart, first_raw] at h |
| exact h |
|
|
| theorem secondFoot_projection_finite (δ r : ℝ) {S : Set (ℝ × ℝ)} (hS : IsCompact S) : |
| Measure.euclideanHausdorffMeasure 2 |
| (projection (secondAxis δ r) '' (secondFoot '' S)) ≠ ⊤ := by |
| have hp : inner ℝ normalSecond (velocitySecond δ r) = 0 := by |
| simp [inner_coord, normalSecond, velocitySecond] |
| have h := RadialProjection.affine_projection_finite originSecond |
| (unit (normalSecond + velocitySecond δ r)) radialSecond transverseSecond |
| (unit_norm (RadialProjection.sideways_ne_zero normalSecond_unit hp)) hS |
| rw [second_chart, second_raw] at h |
| exact h |
|
|
| theorem firstBase_area {δ : ℝ} (hδ : 0 ≤ δ) {R : ℝ → ℝ} |
| (hR : ContinuousOn R (Icc (-1) 1)) (hR0 : ∀ r ∈ Icc (-1) 1, 0 ≤ R r) |
| (N : ℕ) (i : Fin N) : area (firstBase δ R N i) = localArea δ R N i := by |
| have hM : 0 ≤ maxCap R N i := RadialRiemann.capMax_nonneg (by norm_num) hR hR0 i |
| have hρ : 0 ≤ halfwidth N := by unfold halfwidth; positivity |
| have he : 0 ≤ collar δ N := by unfold collar; positivity |
| rw [firstBase, firstFoot_projection_area, foot, RadialFootprints.halfheightFootprint_area _ hρ he hM] |
| unfold localArea |
| ring |
|
|
| theorem secondBase_area {δ : ℝ} (hδ : 0 ≤ δ) {R : ℝ → ℝ} |
| (hR : ContinuousOn R (Icc (-1) 1)) (hR0 : ∀ r ∈ Icc (-1) 1, 0 ≤ R r) |
| (N : ℕ) (i : Fin N) : area (secondBase δ R N i) = localArea δ R N i := by |
| have hM : 0 ≤ maxCap R N i := RadialRiemann.capMax_nonneg (by norm_num) hR hR0 i |
| have hρ : 0 ≤ halfwidth N := by unfold halfwidth; positivity |
| have he : 0 ≤ collar δ N := by unfold collar; positivity |
| rw [secondBase, secondFoot_projection_area, foot, RadialFootprints.halfheightFootprint_area _ hρ he hM] |
| unfold localArea |
| ring |
|
|
| theorem totalArea_eq_two_sum {δ : ℝ} (hδ : 0 ≤ δ) {R : ℝ → ℝ} |
| (hR : ContinuousOn R (Icc (-1) 1)) (hR0 : ∀ r ∈ Icc (-1) 1, 0 ≤ R r) |
| (N : ℕ) : totalArea δ R N = 2 * ∑ i : Fin N, localArea δ R N i := by |
| unfold totalArea |
| simp_rw [firstBase_area hδ hR hR0, secondBase_area hδ hR hR0] |
| ring |
|
|
| end CylinderCovering.HalfheightFinite |
|
|
| end |
|
|
| end OAI |
|
|
