Equation of Cornu Spiral/Parametric

Theorem
Let $K$ be a Cornu spiral embedded in a Cartesian coordinate plane such that the origin coincides with the point at which $s = 0$.

Then $K$ can be expressed by the parametric equations:
 * $\begin {cases} x = a \sqrt 2 \map {\operatorname C} {\dfrac s {a \sqrt 2} } \\ y = a \sqrt 2 \map {\operatorname S} {\dfrac s {a \sqrt 2} } \end {cases}$

where:
 * $\operatorname C$ denotes the Fresnel cosine integral function
 * $\operatorname S$ denotes the Fresnel sine integral function.