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.

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 inconsistent If 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): spiral
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): spiral