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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 44 45 46 47 48 49 | 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
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