Clothoid

\[x(s) = \int_0^s \cos(\lambda t^2) \mathrm{d}t = \frac{1}{\sqrt{\lambda}} \int_0^{\sqrt{\lambda}s} \cos(t^2) \mathrm{d}t \\ y(s) = \int_0^s \sin(\lambda t^2) \mathrm{d}t = \frac{1}{\sqrt{\lambda}} \int_0^{\sqrt{\lambda}s} \sin(t^2) \mathrm{d}t\]

\[l(s) = s \\ \frac{\mathrm{d}l}{\mathrm{d}s} = 1 \\ \kappa(s) = 2 \lambda s \\ \frac{\mathrm{d} \kappa}{\mathrm{d} s} = 2 \lambda \\ \theta(s) = \lambda s^2\]

\[x(s) = \int_0^s \cos(\lambda t^2) \mathrm{d}t = \sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{(2n)!(4n+1)} = x - \frac{1}{10} x^5 + \frac{1}{216} x^9 - \frac{1}{9360} x^{13} + \frac{1}{685440} x^{17} - \frac{1}{76204800} x^{21} + \frac{1}{11975040000} x^{25} - \frac{1}{2528170444800} x^{29} + \dots \\ y(s) = \int_0^s \sin(\lambda t^2) \mathrm{d}t = \sum_{n=0}^\infty (-1)^n \frac{x^{4n+3}}{(2n+1)!(4n+3)} = \frac{1}{3} x^3 - \frac{1}{42} x^7 + \frac{1}{1320} x^{11} - \frac{1}{75600} x^{15} + \frac{1}{6894720} x^{19} + \frac{1}{918086400} x^{23} + \frac{1}{168129561600} x^{27} + \frac{1}{40537905408000} x^{31} + \dots\]

\[x(s) = \int_0^s \cos(\lambda t^2) \mathrm{d}t = \int_0^s \cos(\theta \frac{t^2}{s^2}) \mathrm{d}t = \frac{1}{\sqrt{\theta}} s \int_0^\sqrt{\theta} \cos(t^2) \mathrm{d}t = \frac{1}{\sqrt{\theta}} s \mathrm{F}_{\cos}(\sqrt{\theta}) \\ y(s) = \int_0^s \sin(\lambda t^2) \mathrm{d}t = \int_0^s \sin(\theta \frac{t^2}{s^2}) \mathrm{d}t = \frac{1}{\sqrt{\theta}} s \int_0^\sqrt{\theta} \sin(t^2) \mathrm{d}t = \frac{1}{\sqrt{\theta}} s \mathrm{F}_{\sin}(\sqrt{\theta})\]

\[d = \frac{y}{\cos{\theta}} = s \frac{\mathrm{F}_{\sin}(\sqrt{\theta})}{\sqrt{\theta} \cos{\theta}}\]

\[s = d \frac{\sqrt{\theta} \cos{\theta}}{\mathrm{F}_{\sin}(\sqrt{\theta})}\]

\[\mathrm{shift} = d (\sin \theta + \cos \theta \frac{\mathrm{F}_{\cos}(\sqrt{\theta})}{\mathrm{F}_{\sin}(\sqrt{\theta})})\]