# 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

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**spiral** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**spiral**

- Weisstein, Eric W. "Cardioid." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/Cardioid.html