A plot in the Complex Plane of the points

(1) |

The Slope of the Cornu spiral

(2) |

The Slope of the curve's Tangent Vector (above right figure) is

(3) |

The Cesàro Equation for a Cornu spiral is , where is the Radius of Curvature and the Arc Length. The Torsion is .

Gray (1993) defines a generalization of the Cornu spiral given by parametric equations

(4) | |||

(5) |

The Arc Length, Curvature, and Tangential Angle of this curve are

(6) | |||

(7) | |||

(8) |

The Cesàro Equation is

(9) |

Dillen (1990) describes a class of ``polynomial spirals'' for which the Curvature is a polynomial function of the Arc Length. These spirals are a further generalization of the Cornu spiral.

**References**

Dillen, F. ``The Classification of Hypersurfaces of a Euclidean Space with Parallel Higher Fundamental Form.''
*Math. Z.* **203**, 635-643, 1990.

Gray, A. ``Clothoids.'' §3.6 in *Modern Differential Geometry of Curves and Surfaces.*
Boca Raton, FL: CRC Press, pp. 50-52, 1993.

Lawrence, J. D. *A Catalog of Special Plane Curves.* New York: Dover, pp. 190-191, 1972.

© 1996-9

1999-05-25