The area enclosed by a close plane curve $\displaystyle {\bf x}$ is:

$\displaystyle A=-\frac{1}{2}\oint ds\,{\bf x}\cdot{\bf n}$

where $\displaystyle {\bf n}$ is the principal normal vector of the curve.

Is the converse also true? Meaning, if the previous equation holds, does this imply that $\displaystyle {\bf x}$ is planar?

thanks,
-R