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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 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 | import OAI.Geometry.CylinderCovering.Physical.FiniteDefinitions
import OAI.Geometry.CylinderCovering.Radial.PairedSweepDefinitions
import OAI.Geometry.CylinderCovering.Convexity.CylinderSimilarity
namespace OAI
open Set MeasureTheory
open scoped InnerProductSpace
noncomputable section
namespace RadialPhysical
open CylinderCovering
lemma upperPhysicalAxis_canonical {b : ℝ} (hb : b ≠ 0) (eta s : ℝ) :
upperPhysicalAxis b eta s =
RadialPairedSweep.upperAxis (Real.sqrt 2) (effectiveTilt b eta) (s/b) := by
have hv : point b 1 (-eta*(s^2+b^2)/2) (eta*s) =
RadialPairedSweep.point 1
(-(effectiveTilt b eta)*RadialCaps.G (s/b))
(Real.sqrt 2*effectiveTilt b eta*(s/b)) := by
ext i
fin_cases i <;>
simp [point, RadialPairedSweep.point, height, effectiveTilt, RadialCaps.G] <;>
field_simp
ring
unfold upperPhysicalAxis RadialPairedSweep.upperAxis
rw [hv]
lemma lowerPhysicalAxis_canonical {b : ℝ} (hb : b ≠ 0) (eta s : ℝ) :
lowerPhysicalAxis b eta s =
RadialPairedSweep.lowerAxis (Real.sqrt 2) (effectiveTilt b eta) (s/b) := by
have hv : point b (eta*(s^2+b^2)/2) 1 (eta*s) =
RadialPairedSweep.point ((effectiveTilt b eta)*RadialCaps.G (s/b)) 1
(Real.sqrt 2*effectiveTilt b eta*(s/b)) := by
ext i
fin_cases i <;>
simp [point, RadialPairedSweep.point, height, effectiveTilt, RadialCaps.G] <;>
field_simp
ring
unfold lowerPhysicalAxis RadialPairedSweep.lowerAxis
rw [hv]
lemma mapToCanonical_eq_similarity (b : ℝ) :
mapToCanonical b = Similarity.map (-(!₂[0,0,Real.sqrt 2] : Space))
(2/b) (LinearIsometryEquiv.refl ℝ Space) := by
funext z
change (2/b) • z - (!₂[0,0,Real.sqrt 2] : Space) =
-(!₂[0,0,Real.sqrt 2] : Space) + (2/b) • z
module
lemma mapCenteredToCanonical_eq_similarity (b : ℝ) :
mapCenteredToCanonical b = Similarity.map (0 : Space)
(1/b) (LinearIsometryEquiv.refl ℝ Space) := by
funext z
simp [mapCenteredToCanonical, Similarity.map]
lemma projection_mapToCanonical (b : ℝ) (u z : Space) :
TriangularCovering.projection u (mapToCanonical b z) =
(2/b) • TriangularCovering.projection u z -
TriangularCovering.projection u (!₂[0,0,Real.sqrt 2] : Space) := by
simp only [TriangularCovering.projection, mapToCanonical,
inner_sub_right, inner_smul_right]
module
lemma projection_mapCenteredToCanonical (b : ℝ) (u z : Space) :
TriangularCovering.projection u (mapCenteredToCanonical b z) =
(1/b) • TriangularCovering.projection u z := by
simp only [TriangularCovering.projection, mapCenteredToCanonical, inner_smul_right]
module
lemma projected_image_mapToCanonical (b : ℝ) (u : Space) (S : Set Space) :
TriangularCovering.projection u '' (mapToCanonical b '' S) =
(fun z => (2/b) • z -
TriangularCovering.projection u (!₂[0,0,Real.sqrt 2] : Space)) ''
(TriangularCovering.projection u '' S) := by
simp only [Set.image_image]
congr 1
funext z
exact projection_mapToCanonical b u z
lemma projected_image_mapCenteredToCanonical (b : ℝ) (u : Space) (S : Set Space) :
TriangularCovering.projection u '' (mapCenteredToCanonical b '' S) =
(fun z => (1/b) • z) '' (TriangularCovering.projection u '' S) := by
simp only [Set.image_image]
congr 1
funext z
exact projection_mapCenteredToCanonical b u z
lemma projected_area_mapToCanonical {b : ℝ} (hb : b ≠ 0) (u : Space) (S : Set Space) :
TriangularCovering.area (TriangularCovering.projection u '' (mapToCanonical b '' S)) =
(2/b)^2 * TriangularCovering.area (TriangularCovering.projection u '' S) := by
rw [mapToCanonical_eq_similarity]
exact Similarity.shadowArea_map S _ (div_ne_zero (by norm_num) hb)
(LinearIsometryEquiv.refl ℝ Space) u
lemma projected_area_mapCenteredToCanonical {b : ℝ} (hb : b ≠ 0)
(u : Space) (S : Set Space) :
TriangularCovering.area
(TriangularCovering.projection u '' (mapCenteredToCanonical b '' S)) =
(1/b)^2 * TriangularCovering.area (TriangularCovering.projection u '' S) := by
rw [mapCenteredToCanonical_eq_similarity]
exact Similarity.shadowArea_map S _ (div_ne_zero one_ne_zero hb)
(LinearIsometryEquiv.refl ℝ Space) u
lemma cylinder_mapToCanonical {b : ℝ} (hb : b ≠ 0) (u : Space) (B : Set Space) :
mapToCanonical b '' RadialSweep.cylinder u B =
RadialSweep.cylinder u
((fun z => (2/b) • z - TriangularCovering.projection u
(!₂[0,0,Real.sqrt 2] : Space)) '' B) := by
have hbase : Similarity.transportedBase B u (-(!₂[0,0,Real.sqrt 2] : Space))
(2/b) (LinearIsometryEquiv.refl ℝ Space) =
(fun z => (2/b) • z - TriangularCovering.projection u
(!₂[0,0,Real.sqrt 2] : Space)) '' B := by
unfold Similarity.transportedBase
congr 1
funext z
change (-(!₂[0,0,Real.sqrt 2] : Space) -
(inner ℝ u (-(!₂[0,0,Real.sqrt 2] : Space))) • u) + (2/b) • z =
(2/b) • z - ((!₂[0,0,Real.sqrt 2] : Space) -
(inner ℝ u (!₂[0,0,Real.sqrt 2] : Space)) • u)
simp only [inner_neg_right, neg_smul]
module
rw [mapToCanonical_eq_similarity]
change Similarity.map _ (2/b) (LinearIsometryEquiv.refl ℝ Space) ''
Similarity.carrier B u = Similarity.carrier _ u
rw [Similarity.carrier_image B u _ (div_ne_zero (by norm_num) hb), hbase]
rfl
lemma cylinder_mapCenteredToCanonical {b : ℝ} (hb : b ≠ 0)
(u : Space) (B : Set Space) :
mapCenteredToCanonical b '' RadialSweep.cylinder u B =
RadialSweep.cylinder u ((fun z => (1/b) • z) '' B) := by
rw [mapCenteredToCanonical_eq_similarity]
change Similarity.map _ (1/b) (LinearIsometryEquiv.refl ℝ Space) ''
Similarity.carrier B u = Similarity.carrier _ u
rw [Similarity.carrier_image B u _ (div_ne_zero one_ne_zero hb)]
have hbase : Similarity.baseMap u (0 : Space) (1/b)
(LinearIsometryEquiv.refl ℝ Space) = fun vector => (1/b) • vector := by
funext vector
simp [Similarity.baseMap, Similarity.project]
exact congrArg (fun base => Similarity.carrier (base '' B) u) hbase
end RadialPhysical
end
end OAI
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