Equation of Cornu Spiral/Parametric

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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.


Proof


Sources