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6. Algebraic Curve

An algebraic curve over a field  is an equation , where  is a polynomial in  and  withcoefficients in . A nonsingular algebraic curve is an algebraic curve over  which has no singular points over . A point on an algebraic curve is simply a solution of the equation of the curve. A -rational point is a point  on the curve, where  and  are in the field .

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, folium, hippopede, Kampyle of Eudoxus, Klein quartic, knot curve, lemniscate, limaçon,links curve, pear-shaped curve, piriform, 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, Talbot's curve 8 octic curve pear curve 12 dodecic curve ranunculoid
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