## Area enclosed by a plane curve... and converse

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

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

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

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

thanks,
-R