Math Conventions

A Clothoid for the purposes of this software is any curve for which the curvature is a linear function of arc length. Put another way, the derivative of curvature with respect to arc length must be a constant value. Any such curve can be expressed parametrically in the following form:

\begin{eqnarray*} x(s) &= x_0 + \int_0^s \cos\left(\frac{\dot{\kappa}}{2}s^2 + \kappa_0 s + t_0\right) ds\\ y(s) &= y_0 + \int_0^s \sin\left(\frac{\dot{\kappa}}{2}s^2 + \kappa_0 s + t_0\right) ds\\ \end{eqnarray*}

With curvature and tangent angle described by:

\begin{eqnarray*} \kappa(s) &=& \dot{\kappa}s + \kappa_0\\ t(s) &=& \frac{\dot{\kappa}}{2}s^2 + \kappa_0 s + t_0\\ \end{eqnarray*}

Where each math symbol is mapped to a name in the software and a description according to the following table:

Nomenclature for Clothoid Properties

Math Symbol

Code Symbol

Description

x_0

x0

initial X coordinate

y_0

y0

initial Y coordinate

s

s

arc length

\dot{\kappa}

kd

derivative of curvature

\kappa_0

k0

initial curvature

t_0

t0

initial tangent angle