info prev up next book cdrom email home

Conchoid of Nicomedes

\begin{figure}\begin{center}\BoxedEPSF{conchoid.epsf scaled 705}\end{center}\end{figure}

A curve studied by the Greek mathematician Nicomedes in about 200 BC , also called the Cochloid. It is the Locus of points a fixed distance away from a line as measured along a line from the Focus point (MacTutor Archive). Nicomedes recognized the three distinct forms seen in this family. This curve was a favorite with 17th century mathematicians and could be used to solve the problems of Cube Duplication and Angle Trisection.

In Polar Coordinates,

\end{displaymath} (1)

In Cartesian Coordinates,
\end{displaymath} (2)

The conchoid has $x = a$ as an asymptote and the Area between either branch and the Asymptote is infinite. The Area of the loop is

A = a\sqrt{b^2-a^2} - 2ab\ln\left({b+\sqrt{b^2-a^2}\over a}\right)+ b^2\cos^{-1}\left({a\over b}\right).
\end{displaymath} (3)

See also Conchoid


Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 135-139, 1972.

Lee, X. ``Conchoid of Nicomedes.''

MacTutor History of Mathematics Archive. ``Conchoid.''

Pappas, T. ``Conchoid of Nicomedes.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 94-95, 1989.

Yates, R. C. ``Conchoid.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 31-33, 1952.

© 1996-9 Eric W. Weisstein