Algebraic curves

thumb|right|upright=1.36|Part of a parabola (blue), with various features (other colours). The complete parabola has no endpoints. In this orientation, it extends infinitely to the left, right, and upward. thumb|The parabola is a member of the family of conic sections.

curve obtained by intersecting a cone and a plane

thumb|A hyperbola is an open curve with two branches, the intersection of a plane (geometry)|plane with both halves of a double cone. The plane does not have to be parallel to the axis of the cone; the hyperbola will be symmetrical in any case.|alt=The image shows a double cone in which a geometrical plane has sliced off parts of the top and bottom half; the boundary curve of the slice on the cone is the hyperbola. A double cone consists of two cones stacked point-to-point and sharing the same axis of rotation; it may be generated by rotating a line about an axis that passes through a point of

thumb|500px|The red curve is an epicycloid traced as the small circle (radius rolls around the outside of the large circle (radius . In geometry, an epicycloid (also called hypercycloid) is a plane curve produced by tracing the path of a chosen point on the circumference of a circle—called an epicycle—which rolls without slipping around a fixed circle. It is a particular kind of roulette.

topological property

algebraic variety of dimension one

theorem in mathematics

point on a curve where a moving point on the curve must start to move backward

where the curve is not given by a smooth embedding of a parameter

class of mathematical functions

[[File:Allgemeine zissoide_english.svg|thumb|upright=1.5|

projective abelian algebraic group

mathematical concept

a function related to the indefinite integral of a differential of the first kind

thumb|right|An acnode at the origin (curve described in text)

one-dimensional projective space

family of curves of the form r^n = a^n cos(nθ)

algebraic curve that is a ramified double cover of the projective line

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Result concerning properties of Galois representations associated with modular forms

right|thumb|Brass superegg by Piet Hein (Denmark)|Piet Hein. thumb|Piet Hein Superegg

algebraic curve constructed as a quotient of the complex upper half-plane by the action of a congruence subgroup of the modular group of integral 2×2 matrices

statement about cubic curves in the projective plane

power series with rational exponents

mathematical concept

thumb|500px|Bifolium with A bifolium is a quartic plane curve with equation in Cartesian coordinates:

thumb|right|300px|A crunode at the origin of the curve defined by y^2 - x^2(x+1)=0.

On linear differential equations with certain properties

right|frame|The Trott curve (black) has 28 real bitangents (red). This image shows 7 of them; the others are symmetric with respect to 90° rotations through the origin and reflections through the two blue axes.

mathematical formula

moduli space of degree 0 line bundles on a curve; the identity component of the Picard group

the statement that the number of points of intersection of two planar higher-order curves can be greater than the number of arbitrary points usually needed to define one such curve

curve that is stable in the sense of geometric invariant theory; one with only “mild” singularities and whose automorphism group is finite, such that it can be in the moduli space of curves

connects non-singular algebraic curves with compact Riemann surfaces

thumb|right|300px|Examples of 3-ellipses for three given foci. The progression of the distances is not linear.

kind of mathematical linear code

type of elliptic curve

thumb|right|214px|A tacnode at the origin of the curve defined by (x^2+y^2-3x)^2 - 4x^2(2-x) = 0.