| import OAI.Geometry.CylinderCovering.Radial.Projection |
| import OAI.Geometry.RuledCovering.Main |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory |
| open scoped InnerProductSpace |
|
|
| namespace RadialProjection |
| open TriangularCovering |
|
|
| def coordinateLinear2 (a b : Space) : RuledApproximation.Plane →ₗ[ℝ] Space where |
| toFun z := z 0 • a + z 1 • b |
| map_add' z q := by simp only [PiLp.add_apply, add_smul]; module |
| map_smul' c z := by |
| simp only [PiLp.smul_apply, smul_add, smul_smul, RingHom.id_apply] |
| rfl |
|
|
| def projectedCoordinateLinear (u a b : Space) : RuledApproximation.Plane →ₗ[ℝ] Space := |
| (projLinear u).comp (coordinateLinear2 a b) |
|
|
| @[simp] lemma projectedCoordinateLinear_apply (u a b : Space) |
| (z : RuledApproximation.Plane) : |
| projectedCoordinateLinear u a b z = projection u (z 0 • a + z 1 • b) := rfl |
|
|
| lemma projectedCoordinateLinear_normDet {u : Space} (hu : ‖u‖ = 1) (a b : Space) : |
| (projectedCoordinateLinear u a b).normDet = |inner ℝ u (cross3 a b)| := by |
| let f := projectedCoordinateLinear u a b |
| have h0 : f (EuclideanSpace.basisFun (Fin 2) ℝ 0) = projection u a := by |
| simp [f, EuclideanSpace.basisFun_apply] |
| have h1 : f (EuclideanSpace.basisFun (Fin 2) ℝ 1) = projection u b := by |
| simp [f, EuclideanSpace.basisFun_apply] |
| have hd := f.normDet_sq_eq_det_gram (EuclideanSpace.basisFun (Fin 2) ℝ) |
| simp only [RCLike.ofReal_real_eq_id, id_eq, Matrix.det_fin_two, Matrix.gram_apply, |
| h0, h1] at hd |
| rw [real_inner_comm (projection u a) (projection u b)] at hd |
| have hs : f.normDet ^ 2 = (inner ℝ u (cross3 a b)) ^ 2 := by |
| rw [← projected_gram hu] |
| simpa only [pow_two] using hd |
| have hn := f.normDet_nonneg |
| have ha := abs_nonneg (inner ℝ u (cross3 a b)) |
| have hab := sq_abs (inner ℝ u (cross3 a b)) |
| change f.normDet = _ |
| nlinarith |
|
|
| lemma affine_projected_image (p₀ u a b : Space) (S : Set (ℝ × ℝ)) : |
| projection u '' (coordinateAffine p₀ a b '' S) = |
| (fun x : Space => projection u p₀ + x) '' |
| (projectedCoordinateLinear u a b '' (RuledApproximation.pairPlaneEquiv '' S)) := by |
| simp only [image_image] |
| congr 1 |
| funext z |
| change projection u (p₀ + z.1 • a + z.2 • b) = |
| projection u p₀ + projection u (z.1 • a + z.2 • b) |
| simpa only [projLinear_apply, add_assoc] using (projLinear u).map_add p₀ (z.1 • a + z.2 • b) |
|
|
| theorem affine_projection_measure (p₀ u a b : Space) (hu : ‖u‖ = 1) |
| (S : Set (ℝ × ℝ)) : |
| Measure.euclideanHausdorffMeasure 2 (projection u '' (coordinateAffine p₀ a b '' S)) = |
| ENNReal.ofReal |inner ℝ u (cross3 a b)| * volume S := by |
| rw [affine_projected_image, |
| (isometry_add_left (projection u p₀)).euclideanHausdorffMeasure_image] |
| have hm := (projectedCoordinateLinear u a b).euclideanHausdorffMeasure_image_eq_normDet_mul_volume |
| (RuledApproximation.pairPlaneEquiv '' S) |
| simp only [finrank_euclideanSpace_fin] at hm |
| rw [hm, projectedCoordinateLinear_normDet hu, |
| measurePreserving_image RuledApproximation.pairPlane_measurePreserving |
| RuledApproximation.pairPlaneEquiv.measurableEmbedding] |
|
|
| theorem affine_projection_area (p₀ u a b : Space) (hu : ‖u‖ = 1) |
| (S : Set (ℝ × ℝ)) : |
| area (projection u '' (coordinateAffine p₀ a b '' S)) = |
| |inner ℝ u (cross3 a b)| * volume.real S := by |
| unfold area |
| rw [affine_projection_measure p₀ u a b hu S, ENNReal.toReal_mul, |
| ENNReal.toReal_ofReal (abs_nonneg _)] |
| rfl |
|
|
| theorem radial_affine_projection_area (p₀ e a b w : Space) |
| (he : ‖e‖ = 1) (ha : inner ℝ e a = 0) (hb : inner ℝ e b = 0) |
| (hw : inner ℝ e w = 0) (S : Set (ℝ × ℝ)) : |
| area (projection (unit (e + w)) '' (coordinateAffine p₀ a b '' S)) = |
| Real.sqrt (inner ℝ a a * inner ℝ b b - (inner ℝ a b) ^ 2) / |
| Real.sqrt (1 + ‖w‖ ^ 2) * volume.real S := by |
| rw [affine_projection_area p₀ (unit (e + w)) a b |
| (unit_norm (sideways_ne_zero he hw)), normal_projection_factor he ha hb hw] |
|
|
| theorem affine_projection_finite (p₀ u a b : Space) (hu : ‖u‖ = 1) |
| {S : Set (ℝ × ℝ)} (hS : IsCompact S) : |
| Measure.euclideanHausdorffMeasure 2 (projection u '' (coordinateAffine p₀ a b '' S)) ≠ ⊤ := by |
| rw [affine_projection_measure p₀ u a b hu S] |
| exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top hS.measure_ne_top |
|
|
| theorem affine_projection_isCompact (p₀ u a b : Space) |
| {S : Set (ℝ × ℝ)} (hS : IsCompact S) : |
| IsCompact (projection u '' (coordinateAffine p₀ a b '' S)) := by |
| have hc : Continuous (coordinateAffine p₀ a b) := by |
| change Continuous (fun z : ℝ × ℝ => p₀ + z.1 • a + z.2 • b) |
| fun_prop |
| exact (hS.image hc).image (projLinear u).continuous_of_finiteDimensional |
|
|
| end RadialProjection |
|
|
| end |
|
|
| end OAI |
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|
