# 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.

This page needs the help of a knowledgeable authority.In particular: Not sure whether there are scaling factors that need to be applied -- the definition of $\operatorname C$ and $\operatorname S$ within the literature is inconsistentIf you are knowledgeable in this area, then you can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Help}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Proof

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## 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. https://mathworld.wolfram.com/Cardioid.html