| import OAI.Geometry.CylinderCovering.Convexity.NormalFrame |
| import OAI.Geometry.TriangularCovering.Main |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped InnerProductSpace |
| namespace CylinderNormalFrame |
|
|
| theorem shadowArea_eq_frameCost (u : Space) (hu : ‖u‖ = 1) : |
| TriangularCovering.shadowArea u = frameCost u := by |
| rw [TriangularCovering.shadowArea_abs_sum hu] |
| unfold frameCost |
| simp only [Fin.sum_univ_four] |
| rw [inner_normal_zero, inner_normal_one, inner_normal_two, inner_normal_three] |
| simp [TriangularCovering.inner_coord, TriangularCovering.faceNormal, |
| TriangularCovering.H, abs_div, abs_of_nonneg (Real.sqrt_nonneg (3 : ℝ))] |
| have h3 : Real.sqrt 3 ≠ 0 := ne_of_gt (Real.sqrt_pos.2 (by norm_num)) |
| field_simp |
| ring_nf |
|
|
| theorem geometric_shadow_lower_bound (u : Space) (hu : ‖u‖ = 1) : |
| Real.sqrt 2 ≤ TriangularCovering.shadowArea u := by |
| rw [shadowArea_eq_frameCost u hu] |
| exact frameCost_lower_bound u hu |
|
|
| theorem geometric_projection_minimum : TriangularCovering.A_min = Real.sqrt 2 := by |
| unfold TriangularCovering.A_min |
| apply le_antisymm |
| · apply csInf_le |
| · exact ⟨Real.sqrt 2, by |
| rintro a ⟨u, hu, rfl⟩ |
| exact geometric_shadow_lower_bound u hu⟩ |
| · exact ⟨edgeDirection, edgeDirection_unit, |
| (shadowArea_eq_frameCost edgeDirection edgeDirection_unit).trans |
| frameCost_edgeDirection |>.symm⟩ |
| · apply le_csInf |
| · exact ⟨Real.sqrt 2, edgeDirection, edgeDirection_unit, |
| (shadowArea_eq_frameCost edgeDirection edgeDirection_unit).trans |
| frameCost_edgeDirection |>.symm⟩ |
| · rintro a ⟨u, hu, rfl⟩ |
| exact geometric_shadow_lower_bound u hu |
|
|
| end CylinderNormalFrame |
|
|
| end |
|
|
| end OAI |
|
|
