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cead56f | 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 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 | import OAI.Geometry.ProjectionVolume.PolytopeFront
import OAI.Geometry.ProjectionVolume.Brightness
import OAI.Geometry.ProjectionVolume.ProjectionReconstruction
namespace OAI
universe uι
open Set MeasureTheory
open scoped RealInnerProductSpace Pointwise
noncomputable section
namespace Paper092.HPolytope
variable {d : ℕ} {ι : Type uι} [Fintype ι] (P : HPolytope d ι)
theorem velocity_neg (u : Euclidean d) (i : ι) :
P.velocity (-u) i = -P.velocity u i := by simp [velocity]
private theorem brightness_eq_filtered_sum (u : Euclidean d) (hu : u ≠ 0) :
brightness P.body u =
∑ i, if P.velocity u i < 0 then P.faceArea i * |⟪u, P.normal i⟫| else 0 := by
classical
rw [P.brightness_eq_front_sum u hu]
simp only [face_brightness]
rw [← Finset.sum_filter]
exact (Finset.sum_subtype (Finset.univ.filter (fun i => P.velocity u i < 0))
(by simp) (fun i => P.faceArea i * |⟪u, P.normal i⟫|)).symm
theorem brightness_eq_sum (u : Euclidean d) :
brightness P.body u = (∑ i, P.faceArea i * |⟪u, P.normal i⟫|) / 2 := by
classical
by_cases hu : u = 0
· simp [hu, brightness]
have hplus := P.brightness_eq_filtered_sum u hu
have hminus := P.brightness_eq_filtered_sum (-u) (neg_ne_zero.mpr hu)
simp only [brightness_neg, velocity_neg, neg_lt_zero, inner_neg_left, abs_neg] at hminus
have hpoint (i : ι) :
(if P.velocity u i < 0 then P.faceArea i * |⟪u, P.normal i⟫| else 0) +
(if 0 < P.velocity u i then P.faceArea i * |⟪u, P.normal i⟫| else 0) =
P.faceArea i * |⟪u, P.normal i⟫| := by
rcases lt_trichotomy (P.velocity u i) 0 with hi | hi | hi
· simp [hi, not_lt_of_gt hi]
· have hinner : ⟪u, P.normal i⟫ = 0 := by
simpa only [velocity, neg_eq_zero, real_inner_comm] using hi
simp [hi, hinner]
· simp [hi, not_lt_of_gt hi]
have hsum := congrArg (fun f : ι → ℝ => ∑ i, f i) (funext hpoint)
simp only [Finset.sum_add_distrib] at hsum
rw [← hplus, ← hminus] at hsum
linarith
theorem faceArea_nonneg (i : ι) : 0 ≤ P.faceArea i := ENNReal.toReal_nonneg
theorem facetZonotope_isCompact : IsCompact P.facetZonotope := by
classical
exact Finset.sum_induction _ IsCompact (fun _ _ ha hb => ha.add hb)
isCompact_singleton (fun _ _ => SupportGeometry.isCompact_segment _ _)
theorem facetZonotope_nonempty : P.facetZonotope.Nonempty := by
classical
exact Finset.sum_induction _ Set.Nonempty (fun _ _ ha hb => ha.add hb)
(singleton_nonempty 0) (fun _ _ => ⟨_, left_mem_segment ℝ _ _⟩)
theorem facetZonotope_convex : Convex ℝ P.facetZonotope := by
classical
exact Finset.sum_induction _ (Convex ℝ) (fun _ _ ha hb => ha.add hb)
(convex_singleton 0) (fun _ _ => convex_segment _ _)
theorem facetZonotope_support (u : Euclidean d) :
SupportGeometry.support P.facetZonotope (innerSL ℝ u) =
(∑ i, P.faceArea i * |⟪u, P.normal i⟫|) / 2 := by
classical
unfold facetZonotope
rw [SupportGeometry.support_sum _ _
(fun _ _ => SupportGeometry.isCompact_segment _ _)
(fun _ _ => ⟨_, left_mem_segment ℝ _ _⟩)]
have hseg (i : ι) : SupportGeometry.support
(segment ℝ (-(P.faceArea i / 2) • P.normal i)
((P.faceArea i / 2) • P.normal i)) (innerSL ℝ u) =
P.faceArea i * |⟪u, P.normal i⟫| / 2 := by
have heq : segment ℝ (-(P.faceArea i / 2) • P.normal i)
((P.faceArea i / 2) • P.normal i) =
segment ℝ (-(1 / 2 : ℝ) • (P.faceArea i • P.normal i))
((1 / 2 : ℝ) • (P.faceArea i • P.normal i)) := by
congr 1 <;> simp only [smul_smul] <;> congr 1 <;> ring
rw [heq, SupportGeometry.support_centered_segment]
change |⟪u, P.faceArea i • P.normal i⟫| / 2 = _
simp only [real_inner_smul_right, abs_mul,
abs_of_nonneg (P.faceArea_nonneg i)]
simp only [hseg]
exact (Finset.sum_div _ _ _).symm
theorem projectionBody_eq_facetZonotope : projectionBody P.body = P.facetZonotope :=
Paper092.projectionBody_eq_of_brightness_eq_support P.facetZonotope_isCompact
P.facetZonotope_nonempty P.facetZonotope_convex
(fun u => (P.brightness_eq_sum u).trans (P.facetZonotope_support u).symm)
theorem projectionBody_support (u : Euclidean d) :
SupportGeometry.support (projectionBody P.body) (innerSL ℝ u) = brightness P.body u := by
rw [P.projectionBody_eq_facetZonotope, P.facetZonotope_support, P.brightness_eq_sum]
end Paper092.HPolytope
end
end OAI
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