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695bd70 | 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 | import OAI.Geometry.RuledCovering.Projection
namespace OAI
noncomputable section
open Set MeasureTheory Filter Metric
open scoped Topology ContDiff Pointwise
namespace RuledApproximation
abbrev Coordinates := Fin 2 → ℝ
def frameJacobian (M : Coordinates ≃L[ℝ] Plane) : ℝ :=
|(M.trans (EuclideanSpace.equiv (Fin 2) ℝ)).toContinuousLinearMap.det|
lemma frameJacobian_pos (M : Coordinates ≃L[ℝ] Plane) : 0 < frameJacobian M := by
apply abs_pos.mpr
exact (M.trans (EuclideanSpace.equiv (Fin 2) ℝ)).toLinearEquiv.isUnit_det'.ne_zero
lemma frame_integral_image (M : Coordinates ≃L[ℝ] Plane) {K : Set Coordinates}
(hK : MeasurableSet K) (F : Plane → ℝ) :
(∫ p in M '' K, F p) = frameJacobian M * (∫ x in K, F (M x)) := by
let e := EuclideanSpace.equiv (Fin 2) ℝ
let D := M.trans e
have himage : M '' K = e.symm '' (D '' K) := by
rw [← image_comp]
congr 1
rw [himage]
have hmp : MeasurePreserving e.symm volume volume := PiLp.volume_preserving_toLp (Fin 2)
rw [hmp.setIntegral_image_emb e.symm.toHomeomorph.measurableEmbedding]
rw [integral_image_eq_integral_abs_det_fderiv_smul volume hK
(fun x _ => D.hasFDerivAt.hasFDerivWithinAt) D.injective.injOn]
rw [integral_smul]
rfl
lemma frame_volume_image (M : Coordinates ≃L[ℝ] Plane) {K : Set Coordinates}
(hK : MeasurableSet K) : volume.real (M '' K) = frameJacobian M * volume.real K := by
simpa using frame_integral_image M hK (fun _ => (1 : ℝ))
lemma coordinates_closedBall (b : Coordinates) {a : ℝ} (ha : 0 ≤ a) :
closedBall b a = Set.pi univ (fun i => Icc (b i - a) (b i + a)) := by
ext x
simp only [mem_closedBall, dist_eq_norm, pi_norm_le_iff_of_nonneg ha, Real.norm_eq_abs,
Pi.sub_apply, Set.mem_pi, mem_univ, true_implies, mem_Icc, abs_le]
constructor <;> intro h i <;> obtain ⟨h₁, h₂⟩ := h i <;> constructor <;> linarith
lemma coordinates_closedBall_volume (b : Coordinates) {a : ℝ} (ha : 0 ≤ a) :
volume.real (closedBall b a) = (2 * a) ^ 2 := by
rw [coordinates_closedBall b ha, measureReal_def, volume_pi_pi]
simp only [Real.volume_Icc, add_sub_sub_cancel, Fin.prod_univ_two]
rw [ENNReal.toReal_mul]
rw [ENNReal.toReal_ofReal (by positivity : 0 ≤ a + a)]
ring
lemma coordinates_closedBall_parallelogram (b : Coordinates) {a : ℝ} (ha : 0 < a) :
IsParallelogram (closedBall b a) := by
let u : Coordinates := ![2*a, 0]
let v : Coordinates := ![0, 2*a]
refine ⟨![b 0 - a, b 1 - a], u, v, ?_, ?_⟩
· have heq : (![u, v] : Fin 2 → Coordinates) =
fun i => (2 * a) • Pi.basisFun ℝ (Fin 2) i := by
funext i
ext j
fin_cases i <;> fin_cases j <;> simp [u, v, Pi.basisFun_apply]
rw [heq]
exact (Pi.basisFun ℝ (Fin 2)).linearIndependent.units_smul
(fun _ => Units.mk0 (2 * a) (by positivity))
· rw [coordinates_closedBall b ha.le]
ext x
simp only [Set.mem_pi, mem_univ, true_implies, mem_Icc, Set.mem_ofPred_eq]
constructor
· intro hx
refine ⟨(x 0 - (b 0 - a)) / (2*a), ?_, (x 1 - (b 1 - a)) / (2*a), ?_, ?_⟩
· constructor
· exact div_nonneg (sub_nonneg.mpr (hx 0).1) (by positivity)
· apply (div_le_one (by positivity)).mpr
linarith [(hx 0).2]
· constructor
· exact div_nonneg (sub_nonneg.mpr (hx 1).1) (by positivity)
· apply (div_le_one (by positivity)).mpr
linarith [(hx 1).2]
· ext i
fin_cases i <;> simp [u, v, (by positivity : 2*a ≠ 0)]
· rintro ⟨s, hs, t, ht, rfl⟩ i
fin_cases i <;> simp [u, v] <;> constructor <;> nlinarith [hs.1, hs.2, ht.1, ht.2]
end RuledApproximation
end
end OAI
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