openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Beams /ProjectionFactorDefinitions.lean
| import OAI.Geometry.CylinderCovering.Angular.GeneralDefinitions | |
| import OAI.Geometry.RuledCovering.Model | |
| import OAI.Geometry.CylinderCovering.Beams.sDefinitions | |
| namespace OAI | |
| noncomputable section | |
| open Set MeasureTheory | |
| open scoped InnerProductSpace | |
| namespace ShortenedBeam.ProjectionFactor | |
| open RuledApproximation CylinderCovering | |
| abbrev Plane := RuledApproximation.Plane | |
| def direction (β : ℝ) (v : Plane) : Space := vec3 (β * v 0) (β * v 1) 1 | |
| def axis (β : ℝ) (v : Plane) : Space := ‖direction β v‖⁻¹ • direction β v | |
| def horizontal (β : ℝ) (p : Plane) : Space := vec3 (β * p 0) (β * p 1) 0 | |
| def base (β : ℝ) (v : Plane) (S : Set Plane) : Set Space := | |
| TriangularCovering.projection (axis β v) '' (horizontal β '' S) | |
| def Statement : Prop := | |
| ∀ (v : Plane) (S : Set Plane), IsCompact S → | |
| let β : ℝ := 1 / Real.sqrt 2 | |
| ‖axis β v‖ = 1 ∧ | |
| IsCompact (base β v S) ∧ | |
| Measure.euclideanHausdorffMeasure 2 (base β v S) ≠ ⊤ ∧ | |
| (∀ z ∈ base β v S, inner ℝ (axis β v) z = 0) ∧ | |
| AngularGeneral.carrier (axis β v) (base β v S) = | |
| {z | ∃ p ∈ S, ∃ t : ℝ, z = horizontal β p + t • direction β v} ∧ | |
| TriangularCovering.area (base β v S) / (β ^ 2 / Real.sqrt 2) = | |
| volume.real S * Real.sqrt 2 / Real.sqrt (1 + ‖v‖ ^ 2 / 2) | |
| end ShortenedBeam.ProjectionFactor | |
| end | |
| end OAI | |