| import OAI.Geometry.CylinderCovering.Convexity.GlobalDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.PairedSweepDefinitions |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace CylinderCovering.RadialEqualityCover |
| open Set |
| open RadialPairedSweep |
|
|
| def firstAxis : Space := point 1 0 0 |
| def secondAxis : Space := point 0 1 0 |
|
|
| def firstBase : Set Space := |
| {z | ∃ y t : ℝ, 0 ≤ t ∧ t ≤ 1 ∧ |y| ≤ 1 - t ∧ |
| z = point 0 y (Real.sqrt 2 * t)} |
|
|
| def secondBase : Set Space := |
| (fun z : Space => point (z 1) (z 0) (-z 2)) '' firstBase |
|
|
| def Statement : Prop := |
| ∃ C : Fin 2 → TriangularCovering.TriCylinder, |
| (C 0).axis = firstAxis ∧ (C 1).axis = secondAxis ∧ |
| (C 0).base = firstBase ∧ (C 1).base = secondBase ∧ |
| triangularCovers (tetrahedron (Real.sqrt 2)) C ∧ |
| (∀ i, TriangularCovering.area (C i).base = Real.sqrt 2) ∧ |
| TriangularCovering.totalArea C = 2 * Real.sqrt 2 ∧ |
| TriangularCovering.totalArea C = |
| minimumShadowArea (tetrahedron (Real.sqrt 2)) / 2 ∧ |
| (∀ i, shadowArea (tetrahedron (Real.sqrt 2)) (C i).axis = 4 * Real.sqrt 2) ∧ |
| triangularRelativeCost (tetrahedron (Real.sqrt 2)) C = 1 / 2 |
|
|
| end CylinderCovering.RadialEqualityCover |
|
|
| end |
|
|
| end OAI |
|
|
