Definition:Curvature/Unit-Speed Parametric Form

Definition

Let $C$ be a curve defined by a real function which is twice differentiable.

Let $C$ be embedded in a cartesian plane and defined by the parametric equations:

$\begin{cases} x = \map x t \\ y = \map y t \end{cases}$

Let $C$ have the unit-speed parametrization:

$x'^2 + y'^2 = 1$

The curvature $\kappa$ of $C$ at a point $P = \tuple {x, y}$ is given by:

$\kappa = \sqrt{x^2 + y^2}$

This page needs the help of a knowledgeable authority. In particular: Clearly this is unsigned curvature; the sign is somehow determined by the orientation of the plane and the curve Seemingly, just different styles. Signed curvature is a special for plane curves, and should be called 'signed' curvature. 'Sign' is only useful if the orientation of the parameterization is of interest. 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.

where:

$x' = \dfrac {\d x} {\d t}$ is the derivative of $x$ with respect to $t$ at $P$

$y' = \dfrac {\d y} {\d t}$ is the derivative of $y$ with respect to $t$ at $P$

$x$ and $y$ are the second derivatives of $x$ and $y$ with respect to $t$ at $P$.

Also see

• Equivalence of Definitions of Curvature

• Results about curvature can be found here.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 1$: What Is Curvature? The Euclidean Plane