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 ,
where ,
giving area
The curvature and tangential
angle are given by
See also Conchoid ,
Conchoid
of de Sluze
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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." http://www.jimloy.com/geometry/trisect.htm#curves . MacTutor
History of Mathematics Archive. "Conchoid." http://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
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