| 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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|
