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The parameter $e\ge 0$ in a focus-centred polar coordinates equation $r=\ell /(1-e\cos \theta )$ for a conic section. It distinguishes an ellipse ($e<1$), a parabola ($e=1$), and a hyperbola ($e>1$).

The focal line of an ellipse is the straight line through its two foci.

Constant areal velocity gives constant specific angular momentum $l=r^2\dot{\theta }$, hence zero transverse acceleration and a central force. For a focus-based ellipse $u=1/r=(1+\epsilon \cos \theta )/A$, one has $u_{\theta \theta }+u=1/A$. The Binet equation then gives the displayed radial acceleration, proving an attractive inverse-square law. An orbit's shape alone does not specify its acceleration without a time law.

For a plane inertia-gravity wave on a resting layer, $\eta ={\eta }_0\cos (kx-\omega t)$, the horizontal velocities are $u=U\cos \theta$ and $v=V\sin \theta$, where $U=\omega {\eta }_0/(k{h}_0)$ and $V=f{\eta }_0/(k{h}_0)$. First-order integration at a fixed equilibrium coordinate gives $\xi =-U\sin \theta /\omega$, $\upsilon =V\cos \theta /\omega$. The particle trajectory is an ellipse with axis ratio $|f|/\omega$. For positive $f$, the orbit is clockwise in the $(x,y)$ plane.