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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 | 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
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