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Conchoid of Nicomedes

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ConchoidofNicomedesCurvesConchoid of Nicomedes animationConchoidofNicomedes

A curve with polar coordinates,

r=b+asectheta
(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 0<a/b<1, a/b=1, and a/b>1. (For a=0, 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

(x-a)^2(x^2+y^2)=b^2x^2
(2)

or

(a-b-x)(a+b-x)x^2+(a-x)^2y^2=0.
(3)

The conchoid has x=a as an asymptote, and the area between either branch and the asymptote is infinite.

ConchoidofNicomedesLoop

A conchoid with 0<a/b<1 has a loop for theta in [x,2pi-x], where x=sec^(-1)(-b/a), giving area

A=1/2int_x^(2pi-x)r^2dtheta
(4)
=1/2int_(sec^(-1)(-b/a))^(2pi-sec^(-1)(-b/a))(b+asectheta)^2dtheta
(5)
=asqrt(b^2-a^2)-2abln(b-sqrt(b^2-a^2))+b^2cos^(-1)(a/b).
(6)

The curvature and tangential angle are given by

kappa(t)=(b(b+3asect-2asec^3t))/((b^2+2absect+a^2sec^4t)^(3/2))
(7)
phi(t)=-1/2pi+t+tan^(-1)[((a+bcost)cott)/a].
(8)

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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