| import OAI.Geometry.CylinderCovering.Convexity.NormalFrameDefinitions |
| import OAI.Geometry.TriangularCovering.Basic |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory |
| open scoped InnerProductSpace |
| namespace CylinderBarycentricMetric |
|
|
| abbrev Physical := EuclideanSpace ℝ (Fin 3) |
| abbrev Barycentric := EuclideanSpace ℝ (Fin 4) |
|
|
| def tetrahedron : Set Barycentric := |
| {p | (∀ i, 0 ≤ p i) ∧ ∑ i : Fin 4, p i = 1} |
|
|
| def projection (u p : Barycentric) : Barycentric := p - inner ℝ u p • u |
|
|
| def shadowArea (u : Barycentric) : ℝ := |
| (Measure.euclideanHausdorffMeasure 2 (projection u '' tetrahedron)).toReal |
|
|
| def embedding : Physical →ₗ[ℝ] Barycentric where |
| toFun z := !₂[z 0 / Real.sqrt 2 - z 2 / 2, |
| -z 0 / Real.sqrt 2 - z 2 / 2, |
| z 1 / Real.sqrt 2 + z 2 / 2, |
| -z 1 / Real.sqrt 2 + z 2 / 2] |
| map_add' z w := by |
| ext i |
| fin_cases i <;> simp <;> ring |
| map_smul' r z := by |
| ext i |
| fin_cases i <;> simp <;> ring |
|
|
| def inverse (p : Barycentric) : Physical := |
| !₂[(p 0 - p 1) / Real.sqrt 2, |
| (p 2 - p 3) / Real.sqrt 2, |
| p 2 + p 3] |
|
|
| def offset : Barycentric := !₂[1 / 2, 1 / 2, 0, 0] |
|
|
| def fromPhysical (z : Physical) : Barycentric := |
| offset + (1 / Real.sqrt 2) • embedding z |
|
|
| def ProjectionStatement : Prop := |
| (∀ u : Barycentric, ‖u‖ = 1 → (∑ i : Fin 4, u i = 0) → |
| shadowArea u = (1 / 2 : ℝ) * ∑ i : Fin 4, |u i|) ∧ |
| IsLeast {a : ℝ | ∃ u : Barycentric, |
| ‖u‖ = 1 ∧ ∑ i : Fin 4, u i = 0 ∧ a = shadowArea u} (1 / Real.sqrt 2) |
|
|
| end CylinderBarycentricMetric |
|
|
| end |
|
|
| end OAI |
|
|
