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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 40 41 42 43 | 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
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