Clothoids

A *clothoid* is a curve with linearly varying curvature. Since the intention of this research is to create a track with continuous curvature, clothoids are used extensively. An illustration of a clothoid is shown in figure A.1

The position of points along a clothoid is evaluated with the use of the *Fresnel Integrals*, shown below:

(A.1) | |||

(A.2) |

Here is the implicit variable along the curve.

These integrals cannot be solved analytically. Algorithms, as well as C code, have been created to evaluate the Fresnel Integrals[14].

The equations of a clothoid are:

(A.3) | |||

(A.4) |

In the above, is a scaling parameter and is the implicit variable, in general ranging from zero to . The range of determines the variation of curvature within the clothoid, as well as the initial and final tangent angles.

Some useful parameters of clothoids are given below[15].

**arclength**- The arclength, , of a clothoid at a given value of can be found from the scaling parameter, .

**curvature**- The curvature at a given value of is determined from the following.

**tangent angle**- The tangent angle is found with this equation.

In the methodology being used, the clothoid will always have zero curvature and zero tangent angle at one end, with a specified radius and angle at the other end. The parameter and the range of can be found using these specified values.

(A.8) |

(A.9) |

(A.10) | |||

(A.11) |

The arclength of the clothoid can now be calculated.

(A.12) |