Parallel Curve

The two branches of the parallel curve a distance $k$ away from a parametrically represented curve $(f(t),g(t))$ are

$$x = f\pm {kg'\over\sqrt{f'^2+g'^2}}$$

$$y = g\mp {kf'\over\sqrt{f'^2+g'^2}}.$$

The above figure shows the curves parallel to the Ellipse.

References

Gray, A. ``Parallel Curves.'' §5.7 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 95-97, 1993.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 42-43, 1972.

Lee, X. ``Parallel.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/Parallel_dir/parallel.html.

Yates, R. C. ``Parallel Curves.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 155-159, 1952.