| import OAI.Geometry.CylinderCovering.Radial.Projection | |
| namespace OAI | |
| noncomputable section | |
| open Set | |
| open scoped InnerProductSpace | |
| namespace RadialProjection | |
| open TriangularCovering | |
| theorem transverse_projection_eq_zero_iff {e w z : Space} | |
| (he : ‖e‖ = 1) (hw : inner ℝ e w = 0) (hz : inner ℝ e z = 0) : | |
| projection (unit (e + w)) z = 0 ↔ z = 0 := by | |
| have hnorm : ‖e + w‖ ≠ 0 := norm_ne_zero_iff.mpr (sideways_ne_zero he hw) | |
| have hinner : inner ℝ e (unit (e + w)) = ‖e + w‖⁻¹ := by | |
| simp [TriangularCovering.unit, inner_smul_right, inner_add_right, | |
| he, hw] | |
| constructor | |
| · intro hp | |
| have heq : z = (inner ℝ (unit (e + w)) z) • unit (e + w) := | |
| sub_eq_zero.mp hp | |
| have hmul := congrArg (fun v : Space => inner ℝ e v) heq | |
| rw [hz, inner_smul_right, hinner] at hmul | |
| have hc : inner ℝ (unit (e + w)) z = 0 := | |
| (mul_eq_zero.mp hmul.symm).resolve_right (inv_ne_zero hnorm) | |
| simpa only [hc, zero_smul] using heq | |
| · rintro rfl | |
| simp [projection] | |
| def projectedCoordinateAffine (u p₀ a b : Space) : (ℝ × ℝ) →ᵃ[ℝ] Space := | |
| (projLinear u).toAffineMap.comp (coordinateAffine p₀ a b) | |
| @[simp] theorem projectedCoordinateAffine_apply (u p₀ a b : Space) (p : ℝ × ℝ) : | |
| projectedCoordinateAffine u p₀ a b p = | |
| projection u (p₀ + p.1 • a + p.2 • b) := rfl | |
| theorem coordinateAffine_injective (p₀ : Space) {a b : Space} | |
| (hab : LinearIndependent ℝ (![a,b] : Fin 2 → Space)) : | |
| Function.Injective (coordinateAffine p₀ a b) := by | |
| intro p q hpq | |
| have hsum : p.1 • a + p.2 • b = q.1 • a + q.2 • b := by | |
| change p₀ + p.1 • a + p.2 • b = p₀ + q.1 • a + q.2 • b at hpq | |
| exact add_left_cancel (by simpa only [add_assoc] using hpq) | |
| have hcoords : (![p.1,p.2] : Fin 2 → ℝ) = ![q.1,q.2] := | |
| hab.fintypeLinearCombination_injective (by | |
| simpa [Fintype.linearCombination_apply, Fin.sum_univ_succ] using hsum) | |
| exact Prod.ext (congrFun hcoords 0) (congrFun hcoords 1) | |
| theorem projectedCoordinateAffine_injective (p₀ : Space) {e w a b : Space} | |
| (he : ‖e‖ = 1) (hw : inner ℝ e w = 0) | |
| (ha : inner ℝ e a = 0) (hb : inner ℝ e b = 0) | |
| (hab : LinearIndependent ℝ (![a,b] : Fin 2 → Space)) : | |
| Function.Injective (projectedCoordinateAffine (unit (e + w)) p₀ a b) := by | |
| intro p q hpq | |
| apply coordinateAffine_injective p₀ hab | |
| apply sub_eq_zero.mp | |
| apply (transverse_projection_eq_zero_iff he hw ?_).mp | |
| · change projLinear (unit (e + w)) | |
| (coordinateAffine p₀ a b p - coordinateAffine p₀ a b q) = 0 | |
| rw [map_sub] | |
| exact sub_eq_zero.mpr hpq | |
| · change inner ℝ e | |
| ((p₀ + p.1 • a + p.2 • b) - (p₀ + q.1 • a + q.2 • b)) = 0 | |
| simp only [inner_sub_right, inner_add_right, inner_smul_right, ha, hb, | |
| mul_zero, add_zero, sub_self] | |
| theorem projectedCoordinateAffine_convex_iff (p₀ : Space) {e w a b : Space} | |
| (he : ‖e‖ = 1) (hw : inner ℝ e w = 0) | |
| (ha : inner ℝ e a = 0) (hb : inner ℝ e b = 0) | |
| (hab : LinearIndependent ℝ (![a,b] : Fin 2 → Space)) (S : Set (ℝ × ℝ)) : | |
| Convex ℝ (projectedCoordinateAffine (unit (e + w)) p₀ a b '' S) ↔ Convex ℝ S := by | |
| constructor | |
| · intro h | |
| have hpre := h.affine_preimage (projectedCoordinateAffine (unit (e + w)) p₀ a b) | |
| rwa [Set.preimage_image_eq _ | |
| (projectedCoordinateAffine_injective p₀ he hw ha hb hab)] at hpre | |
| · exact fun h => h.affine_image _ | |
| theorem projectedCoordinateAffine_affineIndependent_iff | |
| (p₀ : Space) {e w a b : Space} | |
| (he : ‖e‖ = 1) (hw : inner ℝ e w = 0) | |
| (ha : inner ℝ e a = 0) (hb : inner ℝ e b = 0) | |
| (hab : LinearIndependent ℝ (![a,b] : Fin 2 → Space)) | |
| {ι : Type*} (p : ι → ℝ × ℝ) : | |
| AffineIndependent ℝ (projectedCoordinateAffine (unit (e + w)) p₀ a b ∘ p) ↔ | |
| AffineIndependent ℝ p := | |
| (projectedCoordinateAffine (unit (e + w)) p₀ a b).affineIndependent_iff | |
| (projectedCoordinateAffine_injective p₀ he hw ha hb hab) | |
| theorem projectedCoordinateAffine_convexHull (u p₀ a b : Space) (S : Set (ℝ × ℝ)) : | |
| projectedCoordinateAffine u p₀ a b '' convexHull ℝ S = | |
| convexHull ℝ (projectedCoordinateAffine u p₀ a b '' S) := | |
| (projectedCoordinateAffine u p₀ a b).image_convexHull S | |
| end RadialProjection | |
| end | |
| end OAI | |