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