Definition:Radius of Curvature/Cartesian Coordinates

Definition

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

Let $C$ be embedded in a cartesian plane.

The radius of curvature of $C$ at a point $P$ can be expressed in cartesian coordinates as:

$\rho = \size {\dfrac {\paren {1 + y'^2}^{3/2} } {y' '} }$

where:

\(\ds y'\)

\(:=\)

\(\ds \dfrac {\d y} {\d x}\)

\(\ds y' '\)

\(:=\)

\(\ds \dfrac {\d^2 y} {\d x^2}\)

Also see

• Results about radius of curvature can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): curvature
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): curvature