| import OAI.Geometry.CylinderCovering.Physical.FiniteDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.PairedSweepDefinitions |
| import OAI.Geometry.CylinderCovering.Convexity.CylinderSimilarity |
|
|
| namespace OAI |
|
|
| open Set MeasureTheory |
| open scoped InnerProductSpace |
| noncomputable section |
| namespace RadialPhysical |
| open CylinderCovering |
|
|
| lemma upperPhysicalAxis_canonical {b : ℝ} (hb : b ≠ 0) (eta s : ℝ) : |
| upperPhysicalAxis b eta s = |
| RadialPairedSweep.upperAxis (Real.sqrt 2) (effectiveTilt b eta) (s/b) := by |
| have hv : point b 1 (-eta*(s^2+b^2)/2) (eta*s) = |
| RadialPairedSweep.point 1 |
| (-(effectiveTilt b eta)*RadialCaps.G (s/b)) |
| (Real.sqrt 2*effectiveTilt b eta*(s/b)) := by |
| ext i |
| fin_cases i <;> |
| simp [point, RadialPairedSweep.point, height, effectiveTilt, RadialCaps.G] <;> |
| field_simp |
| ring |
| unfold upperPhysicalAxis RadialPairedSweep.upperAxis |
| rw [hv] |
|
|
| lemma lowerPhysicalAxis_canonical {b : ℝ} (hb : b ≠ 0) (eta s : ℝ) : |
| lowerPhysicalAxis b eta s = |
| RadialPairedSweep.lowerAxis (Real.sqrt 2) (effectiveTilt b eta) (s/b) := by |
| have hv : point b (eta*(s^2+b^2)/2) 1 (eta*s) = |
| RadialPairedSweep.point ((effectiveTilt b eta)*RadialCaps.G (s/b)) 1 |
| (Real.sqrt 2*effectiveTilt b eta*(s/b)) := by |
| ext i |
| fin_cases i <;> |
| simp [point, RadialPairedSweep.point, height, effectiveTilt, RadialCaps.G] <;> |
| field_simp |
| ring |
| unfold lowerPhysicalAxis RadialPairedSweep.lowerAxis |
| rw [hv] |
|
|
| lemma mapToCanonical_eq_similarity (b : ℝ) : |
| mapToCanonical b = Similarity.map (-(!₂[0,0,Real.sqrt 2] : Space)) |
| (2/b) (LinearIsometryEquiv.refl ℝ Space) := by |
| funext z |
| change (2/b) • z - (!₂[0,0,Real.sqrt 2] : Space) = |
| -(!₂[0,0,Real.sqrt 2] : Space) + (2/b) • z |
| module |
|
|
| lemma mapCenteredToCanonical_eq_similarity (b : ℝ) : |
| mapCenteredToCanonical b = Similarity.map (0 : Space) |
| (1/b) (LinearIsometryEquiv.refl ℝ Space) := by |
| funext z |
| simp [mapCenteredToCanonical, Similarity.map] |
|
|
| lemma projection_mapToCanonical (b : ℝ) (u z : Space) : |
| TriangularCovering.projection u (mapToCanonical b z) = |
| (2/b) • TriangularCovering.projection u z - |
| TriangularCovering.projection u (!₂[0,0,Real.sqrt 2] : Space) := by |
| simp only [TriangularCovering.projection, mapToCanonical, |
| inner_sub_right, inner_smul_right] |
| module |
|
|
| lemma projection_mapCenteredToCanonical (b : ℝ) (u z : Space) : |
| TriangularCovering.projection u (mapCenteredToCanonical b z) = |
| (1/b) • TriangularCovering.projection u z := by |
| simp only [TriangularCovering.projection, mapCenteredToCanonical, inner_smul_right] |
| module |
|
|
| lemma projected_image_mapToCanonical (b : ℝ) (u : Space) (S : Set Space) : |
| TriangularCovering.projection u '' (mapToCanonical b '' S) = |
| (fun z => (2/b) • z - |
| TriangularCovering.projection u (!₂[0,0,Real.sqrt 2] : Space)) '' |
| (TriangularCovering.projection u '' S) := by |
| simp only [Set.image_image] |
| congr 1 |
| funext z |
| exact projection_mapToCanonical b u z |
| |
| lemma projected_image_mapCenteredToCanonical (b : ℝ) (u : Space) (S : Set Space) : |
| TriangularCovering.projection u '' (mapCenteredToCanonical b '' S) = |
| (fun z => (1/b) • z) '' (TriangularCovering.projection u '' S) := by |
| simp only [Set.image_image] |
| congr 1 |
| funext z |
| exact projection_mapCenteredToCanonical b u z |
| |
| lemma projected_area_mapToCanonical {b : ℝ} (hb : b ≠ 0) (u : Space) (S : Set Space) : |
| TriangularCovering.area (TriangularCovering.projection u '' (mapToCanonical b '' S)) = |
| (2/b)^2 * TriangularCovering.area (TriangularCovering.projection u '' S) := by |
| rw [mapToCanonical_eq_similarity] |
| exact Similarity.shadowArea_map S _ (div_ne_zero (by norm_num) hb) |
| (LinearIsometryEquiv.refl ℝ Space) u |
| |
| lemma projected_area_mapCenteredToCanonical {b : ℝ} (hb : b ≠ 0) |
| (u : Space) (S : Set Space) : |
| TriangularCovering.area |
| (TriangularCovering.projection u '' (mapCenteredToCanonical b '' S)) = |
| (1/b)^2 * TriangularCovering.area (TriangularCovering.projection u '' S) := by |
| rw [mapCenteredToCanonical_eq_similarity] |
| exact Similarity.shadowArea_map S _ (div_ne_zero one_ne_zero hb) |
| (LinearIsometryEquiv.refl ℝ Space) u |
| |
| lemma cylinder_mapToCanonical {b : ℝ} (hb : b ≠ 0) (u : Space) (B : Set Space) : |
| mapToCanonical b '' RadialSweep.cylinder u B = |
| RadialSweep.cylinder u |
| ((fun z => (2/b) • z - TriangularCovering.projection u |
| (!₂[0,0,Real.sqrt 2] : Space)) '' B) := by |
| have hbase : Similarity.transportedBase B u (-(!₂[0,0,Real.sqrt 2] : Space)) |
| (2/b) (LinearIsometryEquiv.refl ℝ Space) = |
| (fun z => (2/b) • z - TriangularCovering.projection u |
| (!₂[0,0,Real.sqrt 2] : Space)) '' B := by |
| unfold Similarity.transportedBase |
| congr 1 |
| funext z |
| change (-(!₂[0,0,Real.sqrt 2] : Space) - |
| (inner ℝ u (-(!₂[0,0,Real.sqrt 2] : Space))) • u) + (2/b) • z = |
| (2/b) • z - ((!₂[0,0,Real.sqrt 2] : Space) - |
| (inner ℝ u (!₂[0,0,Real.sqrt 2] : Space)) • u) |
| simp only [inner_neg_right, neg_smul] |
| module |
| rw [mapToCanonical_eq_similarity] |
| change Similarity.map _ (2/b) (LinearIsometryEquiv.refl ℝ Space) '' |
| Similarity.carrier B u = Similarity.carrier _ u |
| rw [Similarity.carrier_image B u _ (div_ne_zero (by norm_num) hb), hbase] |
| rfl |
|
|
| lemma cylinder_mapCenteredToCanonical {b : ℝ} (hb : b ≠ 0) |
| (u : Space) (B : Set Space) : |
| mapCenteredToCanonical b '' RadialSweep.cylinder u B = |
| RadialSweep.cylinder u ((fun z => (1/b) • z) '' B) := by |
| rw [mapCenteredToCanonical_eq_similarity] |
| change Similarity.map _ (1/b) (LinearIsometryEquiv.refl ℝ Space) '' |
| Similarity.carrier B u = Similarity.carrier _ u |
| rw [Similarity.carrier_image B u _ (div_ne_zero one_ne_zero hb)] |
| have hbase : Similarity.baseMap u (0 : Space) (1/b) |
| (LinearIsometryEquiv.refl ℝ Space) = fun vector => (1/b) • vector := by |
| funext vector |
| simp [Similarity.baseMap, Similarity.project] |
| exact congrArg (fun base => Similarity.carrier (base '' B) u) hbase |
| |
| end RadialPhysical |
| |
| end |
| |
| end OAI |
| |
