| import OAI.Geometry.CylinderCovering.Footprints.Polygon | |
| import OAI.Geometry.CylinderCovering.Radial.ProjectionShapes | |
| namespace OAI | |
| noncomputable section | |
| open Set | |
| open scoped InnerProductSpace | |
| namespace RadialProjection | |
| open TriangularCovering CylinderCovering.Footprints | |
| theorem projected_absolute_polygon (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)) | |
| {m M d A B : ℝ} (hM : 0 ≤ M) (hd : 0 < d) (hA : 0 < A) (hB : 0 < B) : | |
| let f := projectedCoordinateAffine (unit (e+w)) p₀ a b | |
| f '' absoluteFootprint m M d A B = | |
| convexHull ℝ (range (f ∘ positiveAbsoluteVertices m M d A B)) ∪ | |
| convexHull ℝ (range (f ∘ negativeAbsoluteVertices m d A B)) ∧ | |
| Function.Injective (f ∘ positiveAbsoluteVertices m M d A B) ∧ | |
| Function.Injective (f ∘ negativeAbsoluteVertices m d A B) ∧ | |
| ¬ Convex ℝ (f '' absoluteFootprint m M d A B) := by | |
| dsimp only | |
| have hf := projectedCoordinateAffine_injective p₀ he hw ha hb hab | |
| refine ⟨absoluteFootprint_image_eq_union_hulls _ hM hd hA hB, | |
| hf.comp (positiveAbsoluteVertices_injective hM hd hA hB), | |
| hf.comp (negativeAbsoluteVertices_injective hd hA hB), ?_⟩ | |
| intro hconv | |
| exact (absoluteFootprint_not_convex hM hd hA hB.le) | |
| ((projectedCoordinateAffine_convex_iff p₀ he hw ha hb hab _).mp hconv) | |
| end RadialProjection | |
| end | |
| end OAI | |