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