| import OAI.Geometry.CylinderCovering.Footprints.QuadraticFootprintsDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.TriangleDefinitions |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace CylinderCovering.Footprints |
|
|
| def trapezoidVertices (m M d A B : ℝ) : Fin 4 → ℝ × ℝ := |
| let l := -A*d |
| let r := M+A*d |
| ![(l,m*l-(d/2*l+B*d^2)), (l,m*l+(d/2*l+B*d^2)), |
| (r,m*r-(d/2*r+B*d^2)), (r,m*r+(d/2*r+B*d^2))] |
|
|
| def ShapeStatement : Prop := |
| (∀ m M d A B : ℝ, 0 ≤ M → 0 < d → 0 < A → 0 ≤ B → |
| ¬ Convex ℝ (absoluteFootprint m M d A B)) ∧ |
| (∀ m M d A : ℝ, 0 < d → 0 < M+2*A*d → |
| affineFootprint m M d A (A/2) = |
| convexHull ℝ (Set.range (RadialTriangle.vertices (-A*d) (M+A*d) m d)) ∧ |
| AffineIndependent ℝ (RadialTriangle.vertices (-A*d) (M+A*d) m d)) ∧ |
| (∀ m M d A B : ℝ, 0 < d → A/2 < B → 0 < M+2*A*d → |
| affineFootprint m M d A B = convexHull ℝ (Set.range (trapezoidVertices m M d A B)) ∧ |
| Function.Injective (trapezoidVertices m M d A B) ∧ |
| trapezoidVertices m M d A B 1 - trapezoidVertices m M d A B 0 = |
| (0,2*(d/2*(-A*d)+B*d^2)) ∧ |
| trapezoidVertices m M d A B 3 - trapezoidVertices m M d A B 2 = |
| (0,2*(d/2*(M+A*d)+B*d^2))) ∧ |
| (∀ m M d A B : ℝ, 0 ≤ M → 0 < d → 0 ≤ A → A/2 < B → |
| 0 < M+2*A*d → (interior (affineFootprint m M d A B)).Nonempty) |
|
|
| end CylinderCovering.Footprints |
|
|
| end |
|
|
| end OAI |
|
|
