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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 | import OAI.Geometry.ProjectionVolume.ProductArea
import OAI.Geometry.ProjectionVolume.ProductReconstruction
import OAI.Geometry.ProjectionVolume.PolytopeCauchy
namespace OAI
universe uι uκ
noncomputable section
open Set MeasureTheory
open scoped BigOperators RealInnerProductSpace Pointwise
namespace Paper092
namespace HPolytope
variable {r s : ℕ} {ι : Type uι} {κ : Type uκ} [Fintype ι] [Fintype κ]
theorem product_faceArea_inl (hr : 0 < r) (A : HPolytope r ι) (B : HPolytope s κ)
(i : ι) : (A.product B).faceArea (.inl i) =
A.faceArea i * (volume B.body).toReal := by
unfold faceArea
rw [product_face_inl, cartesianBody_eq_preimage,
affineHyperplane_product_area_left hr (A.normal i) (A.normal_unit i) (A.offset i)
(A.face i) (A.face_isCompact i) (fun _ hx => hx.2) B.body B.compact.measurableSet,
ENNReal.toReal_mul]
theorem product_faceArea_inr (hs : 0 < s) (A : HPolytope r ι) (B : HPolytope s κ)
(i : κ) : (A.product B).faceArea (.inr i) =
(volume A.body).toReal * B.faceArea i := by
unfold faceArea
rw [product_face_inr, cartesianBody_eq_preimage,
affineHyperplane_product_area_right hs (B.normal i) (B.normal_unit i) (B.offset i)
(B.face i) (B.face_isCompact i) (fun _ hx => hx.2) A.body A.compact.measurableSet,
ENNReal.toReal_mul]
theorem brightness_product (hr : 0 < r) (hs : 0 < s)
(A : HPolytope r ι) (B : HPolytope s κ) (w : Euclidean (r + s)) :
brightness (cartesianBody A.body B.body) w =
(volume B.body).toReal * brightness A.body (splitEuclideanProduct r s w).1 +
(volume A.body).toReal * brightness B.body (splitEuclideanProduct r s w).2 := by
classical
rw [← product_body A B, (A.product B).brightness_eq_sum,
A.brightness_eq_sum, B.brightness_eq_sum, Fintype.sum_sum_type]
have hleft (i : ι) : ⟪w, (A.product B).normal (.inl i)⟫ =
⟪(splitEuclideanProduct r s w).1, A.normal i⟫ := by
change ⟪w, productLiftLeft r s (A.normal i)⟫ = _
rw [real_inner_comm, inner_productLiftLeft, real_inner_comm]
have hright (i : κ) : ⟪w, (A.product B).normal (.inr i)⟫ =
⟪(splitEuclideanProduct r s w).2, B.normal i⟫ := by
change ⟪w, productLiftRight r s (B.normal i)⟫ = _
rw [real_inner_comm, inner_productLiftRight, real_inner_comm]
simp_rw [product_faceArea_inl hr, product_faceArea_inr hs, hleft, hright]
simp_rw [mul_assoc]
rw [← Finset.mul_sum]
have heq : (∑ i, A.faceArea i * ((volume B.body).toReal *
|⟪(splitEuclideanProduct r s w).1, A.normal i⟫|)) =
(volume B.body).toReal * ∑ i, A.faceArea i *
|⟪(splitEuclideanProduct r s w).1, A.normal i⟫| := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro i _
ring
rw [heq]
ring
end HPolytope
theorem projectionBody_product {r s : ℕ} (hr : 2 ≤ r) (hs : 2 ≤ s)
{ι : Type uι} {κ : Type uκ} [Fintype ι] [Fintype κ] (A : HPolytope r ι) (B : HPolytope s κ) :
projectionBody (cartesianBody A.body B.body) =
cartesianBody ((volume B.body).toReal • projectionBody A.body)
((volume A.body).toReal • projectionBody B.body) :=
projectionBody_cartesian_of_brightness A.body B.body A.volume_pos B.volume_pos
(A.brightness_product (by omega) (by omega) B)
theorem normalizedProjectionVolume_product {r s : ℕ} (hr : 2 ≤ r) (hs : 2 ≤ s)
{ι : Type uι} {κ : Type uκ} [Fintype ι] [Fintype κ] (A : HPolytope r ι) (B : HPolytope s κ) :
normalizedProjectionVolume (cartesianBody A.body B.body) =
normalizedProjectionVolume A.body * normalizedProjectionVolume B.body :=
normalized_cartesian_of_projectionBody_eq (by omega) (by omega) A.body B.body
A.compact B.compact A.volume_pos B.volume_pos (projectionBody_product hr hs A B)
end Paper092
end
end OAI
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