Riccati Parametric Deformations of the Cornu Spiral

Abstract

A parametric deformation of the Cornu spiral is introduced. The parameter is an integration constant which appears in the general solution of the Riccati equation related to the Fresnel integrals. Argand plots of the deformed spirals are presented and a supersymmetric (Darboux) structure of the deformation is revealed through the factorization approach.

Planar motion with Fresnel integrals as components of the velocity

“…These components are plotted in Fig. 2 for three values of K, together with their renowned Argand plot (the positive part of the clothoid/Cornu/Euler spiral [9]) and the speed v(t) = ẋ 2 +ẏ 2 .…”

Planar motion with Fresnel integrals as components of the velocity

“…These components are plotted in Fig. 1 for three values of K, together with their renowned Argand plot (the positive part of the clothoid/Cornu/Euler spiral [9]) and the speed v(t) = ẋ2 + ẏ2 . Integrating again with zero integration constants, the cartesian components for the position are…”

Generalized Cornu-type spirals and their Darboux parametric deformations

“…The purpose of this Letter is to introduce a class of spirals which generalizes the Cornu spiral [1] and to study their properties and supersymmetric deformations using an approach [2] based on parametric Darboux transformations. Like in supersymmetric quantum mechanics, the parameter is the constant of integration of the general Riccati solution [3][4][5][6][7].…”

“…which we call the generalized Fresnel integrals and are parametrized by the arclength of the spiral, s. In principle, n can be any positive integer, but here we will discard the cases n = 0, which is the logarithmic spiral, and n = 1 since the spiral F 1 is the circle. For the case n = 2 and p = π, C 2 (z) and S 2 (z) are the Fresnel integrals and the spiral F 2 is the Cornu spiral that were discussed in [2]. * Electronic address: hcr@ipicyt.edu.mx † Electronic address: mancass@erau.edu ‡ Electronic address: cchsieh@gate.sinica.edu.tw…”