Conchoid of Nicomedes
A curve with polar coordinates,
 |
(1)
|
studied by the Greek mathematician Nicomedes in about 200 BC, also known as 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 for 
, 
, and 
. (For 
, it obviously
degenerates to a circle.)
The conchoid of Nicomedes was a favorite with 17th century mathematicians and could be used to solve the problems of cube duplication,
angle trisection, heptagon
construction, and other Neusis constructions
(Johnson 1975).
In Cartesian coordinates, the conchoid of
Nicomedes may be written
 |
(2)
|
or
 |
(3)
|
The conchoid has 
as an asymptote, and the area
between either branch and the asymptote is infinite.
A conchoid with 
has a loop for ![theta in [x,2pi-x]](/images/equations/ConchoidofNicomedes/Inline7.gif)
,
where 
, giving area
The curvature and tangential
angle are given by
SEE ALSO: Conchoid,
Conchoid
of de Sluze
REFERENCES:
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 215,
1987.
Johnson, C. "A Construction for a Regular Heptagon." Math. Gaz. 59,
17-21, 1975.
Lawrence, J. D. A
Catalog of Special Plane Curves. New York: Dover, pp. 135-139, 1972.
Loomis, E. S. "The Conchoid." §2.2 in The Pythagorean Proposition: Its Demonstrations Analyzed and Classified and Bibliography
of Sources for Data of the Four Kinds of "Proofs," 2nd ed. Reston,
VA: National Council of Teachers of Mathematics, pp. 20-22, 1968.
Loy, J. "Trisection of an Angle." https://www.jimloy.com/geometry/trisect.htm#curves.
MacTutor History of Mathematics Archive. "Conchoid." https://www-groups.dcs.st-and.ac.uk/~history/Curves/Conchoid.html.
Pappas, T. "Conchoid of Nicomedes." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 94-95,
1989.
Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New
York: Dover, p. 327, 1958.
Steinhaus, H. Mathematical
Snapshots, 3rd ed. New York: Dover, pp. 154-155, 1999.
Szmulowicz, F. "Conchoid of Nicomedes from Reflections and Refractions in a
Cone." Amer. J. Phys. 64, 467-471, Apr. 1996.
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England:
Penguin Books, p. 34, 1986.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
pp. 38-39, 1991.
Yates, R. C. "Conchoid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards,
pp. 31-33, 1952.
CITE THIS AS:
Weisstein, Eric W. "Conchoid of Nicomedes."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConchoidofNicomedes.html
![-1/2pi+t+tan^(-1)[((a+bcost)cott)/a].](/images/equations/ConchoidofNicomedes/Inline23.gif)
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