Module banked 

Evaluation of banked curves: point, tangent and the rolled section frame.

A Banked3 is evaluated by PLAN distance d, the parameter of its centreline. The point is base(d) + e(d) z, e the pivot law. The tangent is that point’s own: its derivative in d is (p, g + e'), p the unit plan tangent, g the profile grade and e' the pivot’s rate, so a pivot that moves (rotation about the low rail through a transition) tilts the tangent by exactly what it adds to the grade. With a constant pivot it is the centreline’s tangent.

A held-rail pivot piece (CantForm::AboutRail, #279) is derived from the cant law: e = e0 + s D / 2, s = +-1. Over a height piece of the cant its derivatives are s D^(k) / 2; over an angle piece, with D = b sin(psi), e' = s (b / 2) cos(psi) psi', e'' = s (b / 2) (cos(psi) psi'' - sin(psi) psi'^2) and e''' = s (b / 2) (cos(psi) psi''' - 3 sin(psi) psi' psi'' - cos(psi) psi'^3), so with P_k = sup |psi^(k)| the certified bounds are (b / 2) P_1, (b / 2) (P_2 + P_1^2) and (b / 2) (P_3 + 3 P_1 P_2 + P_1^3). The Viennese bend’s P_k are exact over the span asked for: each derivative of its shape is taken at the span’s ends and at its critical points inside (viennese_sup); over a whole piece they are 140 / 64, 16.8 / sqrt(5) and 52.5 times |dpsi| / L^k. Where the pivot is derived, the cant law’s seams are the pivot’s too (derived_cant_seams).

The section is rolled about that tangent t. With n the horizontal left normal and u = t x n, the roll rho gives the lateral axis l = cos(rho) n + sin(rho) u (towards the left rail head) and the section up v = -sin(rho) n + cos(rho) u; rho comes from BankConvention: asin(D / b) for a rotation about the tangent, asin(D / (b cos theta)) for an exact vertical rise D. The rail heads are at point +- (b / 2) l. Every refusal of the curve crate’s BankError reaches the caller by name.

Position is as exact as the centreline’s; the frame adds closed-form trigonometry only.

Structs

BankedSection

The section of a banked curve at one station.

Functions

banked_derivative

Derivative of banked_point in plan distance: (p, g + e'). Not unit: its length is sqrt(1 + (g + e')^2).

banked_point

Position on a banked curve at plan distance d: base(d) + e(d) z.

banked_second_derivative

Second derivative of banked_point in plan distance (#252): the centreline’s (p'', z'') plus the pivot’s e'' vertically.

banked_section

The section at plan distance d: rotation point, rolled frame, cant, bank angle and roll.

banked_tangent

Unit tangent of a banked curve at plan distance d.