| import OAI.Geometry.RuledCovering.Main |
| import OAI.Geometry.CylinderCovering.Ruled.ProjectionDefinitions |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory |
| namespace RuledApproximation |
|
|
| def perpendicularProjectionEquiv {h : ℝ} (hh : 0 < h) (g : Plane) : |
| Plane ≃L[ℝ] ((Submodule.span ℝ ({physicalUnit h g} : Set Space))ᗮ) := |
| (LinearEquiv.ofBijective (perpendicularProjection h g) |
| ⟨perpendicularProjection_injective hh g, |
| (LinearMap.injective_iff_surjective_of_finrank_eq_finrank (by |
| simp only [finrank_euclideanSpace_fin, |
| perpendicularPlane_finrank _ (physicalUnit_norm h g)])).mp |
| (perpendicularProjection_injective hh g)⟩).toContinuousLinearEquiv |
|
|
| @[simp] theorem perpendicularProjectionEquiv_apply {h : ℝ} (hh : 0 < h) |
| (g p : Plane) : perpendicularProjectionEquiv hh g p = perpendicularProjection h g p := rfl |
|
|
| lemma projectedCylinder_base_measurable {h : ℝ} (hh : 0 < h) (g : Plane) |
| {T : Set Plane} (hT : MeasurableSet T) : |
| MeasurableSet (projectedCylinder h g T).base := by |
| exact (perpendicularProjectionEquiv hh g).toHomeomorph.measurableEmbedding.measurableSet_image.mpr hT |
|
|
| lemma projectedCylinder_base_bounded {h : ℝ} (hh : 0 < h) (g : Plane) |
| {T : Set Plane} (hT : Bornology.IsBounded T) : |
| Bornology.IsBounded (projectedCylinder h g T).base := by |
| exact (perpendicularProjectionEquiv hh g).toContinuousLinearMap.lipschitzWith.isBounded_image hT |
|
|
| lemma projectedCylinder_base_open {h : ℝ} (hh : 0 < h) (g : Plane) |
| {T : Set Plane} (hT : IsOpen T) : IsOpen (projectedCylinder h g T).base := by |
| exact (perpendicularProjectionEquiv hh g).isOpenMap T hT |
|
|
| lemma projectedCylinder_base_compact {h : ℝ} (hh : 0 < h) (g : Plane) |
| {T : Set Plane} (hT : IsCompact T) : IsCompact (projectedCylinder h g T).base := by |
| exact hT.image (perpendicularProjectionEquiv hh g).continuous |
|
|
| def Cylinder.transport (C : Cylinder) (F : Space ≃ᵢ Space) : Cylinder := |
| (C.rotate F.toRealLinearIsometryEquiv).translate (F 0) |
|
|
| lemma Cylinder.transport_carrier (C : Cylinder) (F : Space ≃ᵢ Space) : |
| (C.transport F).carrier = F '' C.carrier := by |
| rw [Cylinder.transport, Cylinder.translate_carrier, Cylinder.rotate_carrier, ← image_comp] |
| congr 1 |
| funext x |
| simp [IsometryEquiv.toRealLinearIsometryEquiv_apply] |
|
|
| lemma Cylinder.transport_area (C : Cylinder) (F : Space ≃ᵢ Space) : |
| (C.transport F).area = C.area := by |
| rw [Cylinder.transport, Cylinder.translate_area, Cylinder.rotate_area] |
|
|
| lemma Cylinder.transport_parallelogram {C : Cylinder} (hC : C.parallelogram) |
| (F : Space ≃ᵢ Space) : (C.transport F).parallelogram := |
| Cylinder.translate_parallelogram (Cylinder.rotate_parallelogram hC F.toRealLinearIsometryEquiv) (F 0) |
|
|
| def Cylinder.transportBase (C : Cylinder) (F : Space ≃ᵢ Space) : |
| ((Submodule.span ℝ ({C.axis} : Set Space))ᗮ) ≃ᵢ |
| ((Submodule.span ℝ ({(C.transport F).axis} : Set Space))ᗮ) := |
| (rotatedPlane C.axis F.toRealLinearIsometryEquiv).toIsometryEquiv.trans |
| (IsometryEquiv.addLeft ((C.rotate F.toRealLinearIsometryEquiv).transversePart (F 0))) |
|
|
| lemma Cylinder.transport_base (C : Cylinder) (F : Space ≃ᵢ Space) : |
| (C.transport F).base = C.transportBase F '' C.base := by |
| simp only [Cylinder.transport, Cylinder.translate, Cylinder.rotate, |
| Cylinder.transportBase, Set.image_image] |
| rfl |
|
|
| lemma Cylinder.transport_measurable {C : Cylinder} (hC : MeasurableSet C.base) |
| (F : Space ≃ᵢ Space) : MeasurableSet (C.transport F).base := by |
| rw [Cylinder.transport_base] |
| exact (C.transportBase F).toHomeomorph.measurableEmbedding.measurableSet_image.mpr hC |
|
|
| lemma Cylinder.transport_bounded {C : Cylinder} (hC : Bornology.IsBounded C.base) |
| (F : Space ≃ᵢ Space) : Bornology.IsBounded (C.transport F).base := by |
| rw [Cylinder.transport_base] |
| exact (C.transportBase F).isometry.lipschitzWith.isBounded_image hC |
|
|
| lemma Cylinder.transport_open {C : Cylinder} (hC : IsOpen C.base) |
| (F : Space ≃ᵢ Space) : IsOpen (C.transport F).base := by |
| rw [Cylinder.transport_base] |
| exact (C.transportBase F).toHomeomorph.isOpenMap _ hC |
|
|
| lemma Cylinder.transport_compact {C : Cylinder} (hC : IsCompact C.base) |
| (F : Space ≃ᵢ Space) : IsCompact (C.transport F).base := by |
| rw [Cylinder.transport_base] |
| exact hC.image (C.transportBase F).continuous |
|
|
| theorem physicalProjection_proof : PhysicalProjectionStatement := by |
| intro h T g hh hTb hTm F |
| refine ⟨(projectedCylinder h g T).transport F, ?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩ |
| · rw [Cylinder.transport_carrier, projectedCylinder_carrier] |
| · rfl |
| · rw [Cylinder.transport_area, projectedCylinder_area hh] |
| · exact Cylinder.transport_measurable (projectedCylinder_base_measurable hh g hTm) F |
| · exact Cylinder.transport_bounded (projectedCylinder_base_bounded hh g hTb) F |
| · exact (Cylinder.transport_bounded (projectedCylinder_base_bounded hh g hTb) F).measure_lt_top |
| · intro hT |
| exact Cylinder.transport_open (projectedCylinder_base_open hh g hT) F |
| · intro hT |
| exact Cylinder.transport_compact (projectedCylinder_base_compact hh g hT) F |
| · intro hT |
| exact Cylinder.transport_parallelogram (projectedCylinder_parallelogram hh g hT) F |
|
|
| end RuledApproximation |
|
|
| end |
|
|
| end OAI |
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|
