Algebraic Curve (original) (raw)

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An algebraic curve over a field K is an equation f(X,Y)=0, where f(X,Y) is a polynomial in X and Y with coefficients in K. A nonsingular algebraic curve is an algebraic curve over K which has no singular points over K. A point on an algebraic curve is simply a solution of the equation of the curve. A K-rational point is a point(X,Y) on the curve, where X and Y are in the field K.

The following table lists the names of algebraic curves of a given degree.

order curve examples
2 quadratic curve circle, ellipse,hyperbola, parabola
3 cubic curve cissoid of Diocles,conchoid of de Sluze, folium of Descartes, Maclaurin trisectrix, Maltese cross curve, Mordell curve, Ochoa curve, right strophoid, semicubical parabola, serpentine curve, Tschirnhausen cubic, witch of Agnesi
4 quartic curve ampersand curve,bean curve, bicorn, bicuspid curve, bifoliate, bifolium,bitangent-rich curve, bow,bullet nose, butterfly curve, capricornoid, cardioid,Cartesian ovals, Cassini ovals, conchoid of Nicomedes, cruciform,deltoid, devil's curve,Dürer's conchoid, eight curve, fish curve, hippopede,Kampyle of Eudoxus, Kepler's folium, Klein quartic, knot curve, lemniscate, limaçon,links curve, pear-shaped curve, piriform curve, swastika curve, trefoil curve, trifolium
5 quintic curve Burnside curve,butterfly catastrophe curve, stirrup curve
6 sextic curve astroid, atriphtaloid,Cayley's sextic, cornoid,cycloid of Ceva, dumbbell curve, ellipse evolute, epicycloid,Freeth's nephroid, heart curve (first), limaçon evolute, nephroid, quadrifolium,scarabaeus curve, Talbot's curve
8 octic curve pear curve
12 dodecic curve ranunculoid

See also

Algebraic Geometry, Algebraic Variety, Curve

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References

Arbarello, E.; Cornalba, M.; Griffiths, P. A.; and Harris, J. Geometry of Algebraic Curves, I. New York: Springer-Verlag, 1985.Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, 1959.Griffiths, P. A. Introduction to Algebraic Curves. Providence, RI: Amer. Math. Soc., 1989.

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Algebraic Curve

Cite this as:

Weisstein, Eric W. "Algebraic Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AlgebraicCurve.html

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