File size: 6,750 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 | import OAI.Geometry.CylinderCovering.Physical.CoordinatesDefinitions
import OAI.Geometry.CylinderCovering.Radial.CoordinatesDefinitions
namespace OAI
open Set MeasureTheory
noncomputable section
namespace RadialPhysical
structure PhysicalCoordinatesStatement : Prop where
point_distance_squared (b x y t x' y' t' : ℝ) :
‖point b x y t - point b x' y' t'‖ ^ 2 =
(x - x') ^ 2 + (y - y') ^ 2 + 2 * b ^ 2 * (t - t') ^ 2
halfheight_distance {b : ℝ} (hb : 0 < b) (x y t x' y' t' : ℝ) :
dist (canonicalPoint (2 * x / b) (2 * y / b) (2 * t - 1))
(canonicalPoint (2 * x' / b) (2 * y' / b) (2 * t' - 1)) =
(2 / b) * dist (point b x y t) (point b x' y' t')
first_line_dictionary {b : ℝ} (hb : b ≠ 0) (eta x rho s : ℝ) :
canonicalPoint (2 * x / b)
(2 * (rho * s - eta * x * (s ^ 2 + b ^ 2) / 2) / b)
(2 * (1 - rho + eta * x * s) - 1) =
canonicalPoint (2 * x / b)
((2 * rho) * (s / b) - effectiveTilt b eta * (2 * x / b) * ((1 + (s / b) ^ 2) / 2))
(1 - 2 * rho + effectiveTilt b eta * (2 * x / b) * (s / b))
second_line_dictionary {b : ℝ} (hb : b ≠ 0) (eta y rho s : ℝ) :
canonicalPoint (2 * (rho * s + eta * y * (s ^ 2 + b ^ 2) / 2) / b)
(2 * y / b) (2 * (rho + eta * y * s) - 1) =
canonicalPoint ((2 * rho) * (s / b) +
effectiveTilt b eta * (2 * y / b) * ((1 + (s / b) ^ 2) / 2))
(2 * y / b) (-1 + 2 * rho + effectiveTilt b eta * (2 * y / b) * (s / b))
capPolynomial_angular_substitution (b beta r : ℝ) :
capPolynomial b beta (b * r) = b ^ 4 * (canonicalProfile r + beta) / 2
linearCap_dictionary (b eta beta r : ℝ) :
2 * linearCap b eta beta (b * r) =
1 + effectiveTilt b eta ^ 2 * (canonicalProfile r + beta)
squareCap_dictionary (b eta beta r : ℝ) :
2 * squareCap b eta beta (b * r) =
Real.sqrt (1 + 2 * effectiveTilt b eta ^ 2 * (canonicalProfile r + beta))
physical_speed_dictionary (b eta r : ℝ) :
1 + eta ^ 2 * (((b * r) ^ 2 + b ^ 2) ^ 2 / 4 + height b ^ 2 * (b * r) ^ 2) =
1 + effectiveTilt b eta ^ 2 * canonicalSpeed r
scaledCost_dictionary {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ) :
scaledCost b eta T = (b * height b / 2) *
canonicalCost (effectiveTilt b eta) (fun r => 2 * T (b * r))
mapToCanonical_point {b : ℝ} (hb : b ≠ 0) (x y t : ℝ) :
mapToCanonical b (point b x y t) =
canonicalPoint (2 * x / b) (2 * y / b) (2 * t - 1)
mapToCanonical_tetrahedron {b : ℝ} (hb : 0 < b) :
mapToCanonical b '' tetrahedron b = RadialCoordinates.canonicalTetrahedron
scaled_table_rational_rows (tau : ℝ) :
relativeQuadraticTerm 1 tau (1 / 120) = -tau ^ 2 / 60 ∧
relativeQuadraticTerm (1 / 2) (tau / 2) (1 / 120) = -tau ^ 2 / 3840 ∧
relativeQuadraticTerm (1 / 2) tau (4 / 625) = -(77 / 60000) * tau ^ 2 ∧
relativeQuadraticTerm (1 / 2) (2 * tau) (1 / 120) = -tau ^ 2 / 240 ∧
relativeQuadraticTerm 1 (2 * tau) (1 / 200) = -(7 / 75) * tau ^ 2
scaled_cap_half_edge (s : ℝ) :
capPolynomial (1 / 2) (4 / 625) s = s ^ 4 / 4 + s ^ 2 / 16 + 1 / 5000
scaled_table_sqrt_row (tau : ℝ) :
relativeQuadraticTerm (1 / Real.sqrt 2) tau (1 / 125) = -(13 / 3000) * tau ^ 2
scaled_cap_sqrt_edge (s : ℝ) :
capPolynomial (1 / Real.sqrt 2) (1 / 125) s = (s ^ 4 + s ^ 2 / 2) / 4 + 1 / 1000
centered_distance {b : ℝ} (hb : 0 < b) (x y t x' y' t' : ℝ) :
dist (canonicalPoint (x / b) (y / b) t)
(canonicalPoint (x' / b) (y' / b) t') =
(1 / b) * dist (point b x y t) (point b x' y' t')
nonorthPoint_distance_squared (p q t p' q' t' : ℝ) :
‖nonorthPoint p q t - nonorthPoint p' q' t'‖ ^ 2 =
(t - t') ^ 2 + ((p - p') ^ 2 + (q - q') ^ 2) / 2
nonorth_distance (p q t p' q' t' : ℝ) :
dist (canonicalPoint p q t) (canonicalPoint p' q' t') =
Real.sqrt 2 * dist (nonorthPoint p q t) (nonorthPoint p' q' t')
orthonormal_height_dictionary (x y z : ℝ) :
canonicalPoint y x ((1 / Real.sqrt 2) * z) = (!₂[y, x, z] : Space)
orthonormal_cubic_padding (tau H : ℝ) :
(1 / Real.sqrt 2) * (tau ^ 2 * (1 / Real.sqrt 2) * H + 10 * tau ^ 3) =
((1 / Real.sqrt 2) * tau) ^ 2 * H + 20 * ((1 / Real.sqrt 2) * tau) ^ 3
orthonormal_error_margin : 9 * (1 / Real.sqrt 2) ^ 2 < (20 : ℝ)
halfedge_alternative_cap (tau s : ℝ) :
linearCap (1 / 2) (tau / 2) (1 / 120) s =
1 / 2 + tau ^ 2 * ((1 / 2) ^ 2 * s ^ 2 *
((1 / 2) * (1 / 8 + s ^ 2 / 2)) + (1 / 2) ^ 3 / 1920)
halfedge_quadratic_cost (tau : ℝ) :
quadraticCost (1 / 2) tau (4 / 625) =
2 * height (1 / 2) * (1 / 8 - 77 / 480000 * tau ^ 2) ∧
quadraticCost (1 / 2) (tau / 2) (1 / 120) =
height (1 / 2) / 4 - height (1 / 2) * (1 / 2) ^ 4 / 960 * tau ^ 2
centeredLinearCap_dictionary (b eta beta r : ℝ) :
centeredLinearCap b eta beta (b * r) =
1 + effectiveTilt b eta ^ 2 * (canonicalProfile r + beta)
centered_sqrt_cap (eta s : ℝ) :
centeredLinearCap (1 / Real.sqrt 2) eta (1 / 250) s =
1 + eta ^ 2 * ((s ^ 4 + s ^ 2 / 2) / 2 + 1 / 1000)
centered_quadratic_coefficient_neg {b : ℝ} (hb : 0 < b) :
-b / 120 + 2 * b / 1000 < 0
centeredCost_dictionary {b : ℝ} (hb : 0 < b) (eta : ℝ) (R : ℝ → ℝ) :
scaledCost b eta R = 2 * b * height b *
canonicalCost (effectiveTilt b eta) (fun r => R (b * r))
mapCenteredToCanonical_point {b : ℝ} (hb : b ≠ 0) (x y t : ℝ) :
mapCenteredToCanonical b (point b x y t) =
canonicalPoint (x / b) (y / b) t
mapCenteredToCanonical_tetrahedron {b : ℝ} (hb : 0 < b) :
mapCenteredToCanonical b '' centeredTetrahedron b = RadialCoordinates.canonicalTetrahedron
centered_first_line_dictionary {b : ℝ} (hb : b ≠ 0) (eta x rho s : ℝ) :
canonicalPoint (x / b) ((rho * s - eta * x * (s ^ 2 + b ^ 2) / 2) / b)
(1 - rho + eta * x * s) =
canonicalPoint (x / b)
(rho * (s / b) - effectiveTilt b eta * (x / b) * ((1 + (s / b) ^ 2) / 2))
(1 - rho + effectiveTilt b eta * (x / b) * (s / b))
centered_second_line_dictionary {b : ℝ} (hb : b ≠ 0) (eta y rho s : ℝ) :
canonicalPoint ((rho * s + eta * y * (s ^ 2 + b ^ 2) / 2) / b) (y / b)
(-1 + rho + eta * y * s) =
canonicalPoint (rho * (s / b) +
effectiveTilt b eta * (y / b) * ((1 + (s / b) ^ 2) / 2))
(y / b) (-1 + rho + effectiveTilt b eta * (y / b) * (s / b))
centered_fractional_padding {eta : ℝ} (heta : 0 < eta) :
Real.rpow eta (5 / 2) = 4 * Real.sqrt 2 *
Real.rpow (effectiveTilt (1 / Real.sqrt 2) eta) (5 / 2)
end RadialPhysical
end
end OAI
|