Cornu spiral

Description

The Cornu spiral, also called the Euler spiral or clothoid, is the plane curve $t \mapsto (C(t), S(t))$, $t \in \R$, given by the normalised Fresnel integrals $$C(t) = \int_0^t \cos\left(\frac{\pi s^2}{2}\right) ds, \qquad S(t) = \int_0^t \sin\left(\frac{\pi s^2}{2}\right) ds.$$ Its velocity is $(\cos(\pi t^2/2), \sin(\pi t^2/2))$, a unit vector, so $t$ is the arc length measured from the origin. The angle of the tangent is $\pi t^2/2$, so the curvature is $\kappa(t) = \pi t$: it grows linearly with arc length. The curve is symmetric under $t \mapsto -t$, which maps $(C, S)$ to $(-C, -S)$. As $t \to +\infty$ it winds around the point $(1/2, 1/2)$ and converges to it, and as $t \to -\infty$ it converges to $(-1/2, -1/2)$.