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785c727 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 | 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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