An algebraic curve is a one-dimensional algebraic variety . An affine plane algebraic curve over a field locus
defined by an equation polynomial
with coefficients in singular
points over an algebraic closure of point on
the plane curve is a solution of its equation , with
coordinates in extension field . A rational point has both
coordinates in
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 , 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 , lima
bean curve , limaçon , links
curve , Maltese cross 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 ,
Character Variety ,
Curve ,
Etale Morphism ,
p -adic
Section ConjectureExplore with Wolfram|Alpha
References Arbarello, E.; Cornalba, M.; Griffiths, P. A.; and Harris, J. Geometry
of Algebraic Curves, I. Coolidge,
J. L. A
Treatise on Algebraic Plane Curves. Griffiths,
P. A. Introduction
to Algebraic Curves. Krishnan,
S. and Manocha, D. "Numeric-Symbolic Algorithms for Evaluating One-Dimensional
Algebraic Sets." In Proc. 1995 International Symposium on Symbolic and Algebraic
Computation, ISSAC '95 (Ed. A. H. M. Levelt). New York: ACM Press,
pp. 59-67, 1995. https://doi.org/10.1145/220346.220354 . Referenced
on Wolfram|Alpha Algebraic Curve
Cite this as:
Weisstein, Eric W. "Algebraic Curve."
From MathWorld https://mathworld.wolfram.com/AlgebraicCurve.html
Subject classifications
is the locus
defined by an equation 
, where 
is a nonconstant polynomial
with coefficients in 
. A nonsingular algebraic curve has no singular
points over an algebraic closure of 
. A point on
the plane curve is a solution of its equation, with
coordinates in 
or an extension field. A 
-rational point has both
coordinates in 
.
