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curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...

s used in

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, by Wikipedia page. See also

list of curves This is a list of Wikipedia articles about curves used in different fields: mathematics (including geometry, statistics, and applied mathematics), physics, engineering, economics, medicine, biology, psychology, ecology, etc. Mathematics (Geometry) ...

Algebraic curves

Rational curves

Degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement Degree of geographical latitude Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
1

File:FuncionLineal01.svg,

Line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...

Degree 2

File:Circle-withsegments.svg,

Circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

File:Ellipse Properties of Directrix and String Construction.svg,

Ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

File:Parts of Parabola.svg,

Parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...

File:Hyperbola properties.svg,

Hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...

Degree 3

File:CubeChart.svg, Cubic curve File:Polynomialdeg3.svg, Cubic polynomial File:Kartesisches-Blatt.svg, Folium of Descartes File:Cissoide2.svg, Cissoid of Diocles File:Conchoid of deSluze.svg, Conchoid of de Sluze File:Cubic_with_double_point.svg, Cubic with double point File:StrophoidConstruction.svg,

Strophoid In geometry, a strophoid is a curve generated from a given curve and points (the fixed point) and (the pole) as follows: Let be a variable line passing through and intersecting at . Now let and be the two points on whose distance from ...

File:Semicubical_parabola.svg,

Semicubical parabola In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form : y^2 - a^2 x^3 = 0 (with ) in some Cartesian coordinate system. Solving for leads to the ''explicit form'' : y = \ ...

File:Serpentine_curve.png,

Serpentine curve A serpentine curve is a curve whose equation is of the form :x^2y+a^2y-abx=0, \quad ab > 0. Equivalently, it has a parametric representation :x=a\cot(t), y=b\sin (t)\cos(t), or functional representation :y=\frac. The curve has an inflection po ...

File:Trif1111.jpg,

Trident curve In mathematics, a trident curve (also trident of Newton or parabola of Descartes) is any member of the family of curves that have the formula: :xy+ax^3+bx^2+cx=d Trident curves are cubic plane curves with an ordinary double point in the real pr ...

File:MaclaurinTrisectrix.SVG,

Trisectrix of Maclaurin In algebraic geometry, the trisectrix of Maclaurin is a cubic plane curve notable for its trisectrix property, meaning it can be used to trisect an angle. It can be defined as locus of the point of intersection of two lines, each rotating at a un ...

File:CubiqueTschirnhausen.svg, Tschirnhausen cubic File:Witch_of_Agnesi,_a_1,_2,_4,_8.svg, Witch of Agnesi

Degree 4

File:Ampersandcurve.svg,

Ampersand curve In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0, with at least one o ...

File:Bean_curve.svg,

Bean curve In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0, with at least one of ...

File:Bicorn.svg,

Bicorn In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation y^2 \left(a^2 - x^2\right) = \left(x^2 + 2ay - a^2\right)^2. It has two cusps and is symmetric abou ...

File:Bicorn-inf.jpg, Transformed bicorn File:Bicuspid curve.svg, Bicuspid curve File:Bowcurve.svg, Bow curve File:Bullet_nose_curve.svg, Bullet-nose curve File:Cruciform1.png,

Cruciform curve In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree of a polynomial, degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0 ...

File:Deltoid2.gif, Deltoid curve File:Devils_curve_a%3D0.8_b%3D1.svg,

Devil's curve In geometry, a Devil's curve, also known as the Devil on Two Sticks, is a curve defined in the Cartesian plane by an equation of the form : y^2(y^2 - b^2) = x^2(x^2 - a^2). The polar equation of this curve is of the form :r = \sqrt = \sqrt. De ...

File:PedalCurve1.gif,

Hippopede In geometry, a hippopede () is a plane curve determined by an equation of the form :(x^2+y^2)^2=cx^2+dy^2, where it is assumed that and since the remaining cases either reduce to a single point or can be put into the given form with a rotation. ...

File:Kampyle Eudoxus.png, Kampyle of Eudoxus File:Kappa_curve_with_asymptotes_-_by_Pt.png,

Kappa curve In geometry, the kappa curve or Gutschoven's curve is a two-dimensional algebraic curve resembling the Greek letter . The kappa curve was first studied by Gérard van Gutschoven around 1662. In the history of mathematics, it is remembered as one of ...

File:PedalCurve1.gif, Lemniscate of Booth File:Lemniscate-of-Gerono.svg, Lemniscate of Gerono File:Lemniscate_of_Bernoulli.svg, Lemniscate of Bernoulli File:EpitrochoidIn1.gif, Limaçon File:Cardiod_animation.gif, Cardioid File:Lima%C3%A7onTrisectrix.svg,

Limaçon trisectrix In geometry, a limaçon trisectrix is the name for the quartic plane curve that is a trisectrix that is specified as a limaçon. The shape of the limaçon trisectrix can be specified by other curves particularly as a rose, conchoid or epitro ...

File:Quadrifolium.svg,

Quadrifolium The quadrifolium (also known as four-leaved clover) is a type of rose curve with an angular frequency of 2. It has the polar equation: :r = a\cos(2\theta), \, with corresponding algebraic equation :(x^2+y^2)^3 = a^2(x^2-y^2)^2. \, Rotated c ...

(2-

rose A rose is either a woody perennial flowering plant of the genus ''Rosa'' (), in the family Rosaceae (), or the flower it bears. There are over three hundred species and tens of thousands of cultivars. They form a group of plants that can be ...

) File:Spiric section.svg, Spiric sections File:Squircle2.svg, Squircle File:Swastika curve.svg, Swastika curve File:Three-leaved_clover_polar.svg,

Trifolium Curve Clover or trefoil are common names for plants of the genus ''Trifolium'' (from Latin ''tres'' 'three' + ''folium'' 'leaf'), consisting of about 300 species of flowering plants in the legume or pea family Fabaceae originating in Europe. The genus h ...

File:Trott.png,

Trott curve In the theory of algebraic plane curves, a general quartic plane curve has 28 bitangent lines, lines that are tangent to the curve in two places. These lines exist in the complex projective plane, but it is possible to define quartic curves for wh ...

Degree 5

Degree 6

File:HypotrochoidOn4.gif, Astroid File:Atriph05.svg,

Atriphtaloid An atriphtaloid, also called an atriphtothlassic curve, is type of sextic plane curve. It is given by the equation x^4 \left(x^2 + y^2\right) - \left(ax^2 - b\right)^2 = 0, where ''a'' and ''b'' are positive number In mathematics, the sign of ...

File:Neph0b.png,

Nephroid In geometry, a nephroid () is a specific plane curve. It is a type of epicycloid in which the smaller circle's radius differs from the larger by a factor of one-half. Name Although the term ''nephroid'' was used to describe other curves, it was ...

File:Quadrifolium.svg,

Families of variable degree

File:EpitrochoidOn3-generation.gif,

Epicycloid In geometry, an epicycloid 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. Equati ...

File:Epispiral.svg,

Epispiral The epispiral is a plane curve with polar equation :\ r=a \sec. There are ''n'' sections if ''n'' is odd and 2''n'' if ''n'' is even. It is the polar or circle inversion of the rose curve. In astronomy the epispiral is related to the equations ...

File:EpitrochoidIn3.gif, Epitrochoid File:Deltoid2.gif, Hypocycloid File:Lissajous_animation.gif,

Lissajous curve A Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations : x=A\sin(at+\delta),\quad y=B\sin(bt), which describe the superposition of two perpendicular oscillations in x and y dire ...

File:Poinsot1.svg,

Poinsot's spirals In mathematics, Poinsot's spirals are two spirals represented by the polar equations : r = a\ \operatorname (n\theta) : r = a\ \operatorname (n\theta) where csch is the hyperbolic cosecant, and sech is the hyperbolic secant. They are named afte ...

File:7_Petal_rose.svg,

Rose curve A rose is either a woody perennial flowering plant of the genus ''Rosa'' (), in the family Rosaceae (), or the flower it bears. There are over three hundred species and tens of thousands of cultivars. They form a group of plants that can be ...

Curves of genus one

File:Bicuspid_curve.svg, Bicuspid curve File:Line_of_Cassini.svg, Cassini oval File:CubicCurve.svg, Cubic curve File:EllipticCurveCatalog.svg,

Elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...

File:Watt_curve_animated.gif,

Watt's curve In mathematics, Watt's curve is a tricircular plane algebraic curve of degree six. It is generated by two circles of radius ''b'' with centers distance 2''a'' apart (taken to be at (±''a'', 0)). A line segment of length 2''c'' attaches to a p ...

Curves with genus greater than one

File:Butterfly_curve.png, Butterfly curve (algebraic) File:C168.svg,

Elkies trinomial curves Elkies trinomial curve C168 In number theory, the Elkies trinomial curves are certain hyperelliptic curves constructed by Noam Elkies which have the property that rational points on them correspond to trinomial polynomials giving an extension of Q ...

File:Example_of_a_hyperelliptic_curve.svg, Hyperelliptic curve File:Order-7 triangular tiling.svg, Klein quartic File:Modknot11.png,

Classical modular curve In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation :, such that is a point on the curve. Here denotes the -invariant. The curve is sometimes called , though often that notation is used fo ...

Curve families with variable genus

File:Erdos5.png, Erdős lemniscate File:Order-7 triangular tiling.svg,

Hurwitz surface In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(''g'' − 1) automorphisms, where ''g'' is the genus of the surface. This number is maximal by virt ...

File:Mandelcurve2.png, Mandelbrot curve File:Cyc7.png,

Polynomial lemniscate In mathematics, a polynomial lemniscate or ''polynomial level curve'' is a plane algebraic curve of degree 2n, constructed from a polynomial ''p'' with complex coefficients of degree ''n''. For any such polynomial ''p'' and positive real number ' ...

File:Sinusoidal_spirals.svg,

Sinusoidal spiral In algebraic geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates :r^n = a^n \cos(n \theta)\, where is a nonzero constant and is a rational number other than 0. With a rotation about the origin, ...

File:Superellipse_chamfered_square.svg,

Superellipse A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but a different overall shape. In the ...

Transcendental curves

File:Lissajous_curve_5by4.svg,

Bowditch curve A Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations : x=A\sin(at+\delta),\quad y=B\sin(bt), which describe the superposition of two perpendicular oscillations in x and y dir ...

File:Brachistochrone.svg,

Brachistochrone In physics and mathematics, a brachistochrone curve (), or curve of fastest descent, is the one lying on the plane between a point ''A'' and a lower point ''B'', where ''B'' is not directly below ''A'', on which a bead slides frictionlessly under ...

File:Butterfly transcendental curve.svg,

Butterfly curve (transcendental) The butterfly curve is a transcendental plane curve discovered by Temple H. Fay of University of Southern Mississippi in 1989. __TOC__ Equation The curve is given by the following parametric equations: :x = \sin t \!\left(e^ - 2\cos 4t - \ ...

File:Catenary-pm.svg, Catenary File:Clelies_curve.png, Clélies File:Cochleoid.svg,

Cochleoid In geometry, a cochleoid is a snail-shaped curve similar to a strophoid which can be represented by the polar equation :r=\frac, the Cartesian equation :(x^2+y^2)\arctan\frac=ay, or the parametric equations :x=\frac, \quad y=\frac. The cochleoi ...

File:Cycloid_f.gif, Cycloid File:Horopter.png,

Horopter The horopter was originally defined in geometric terms as the locus of points in space that make the same angle at each eye with the fixation point, although more recently in studies of binocular vision it is taken to be the locus of points in spa ...

File:Tautochrone_curve.gif,

Isochrone Isochrone may refer to: * Stellar isochrone, the curve on the Hertzsprung–Russell diagram representing stars of the same age *Isochrone curve, the curve (a cycloid) for which objects starting at different points finish at the same time and point ...

File:Four_point_pursuit_curve.gif,

Pursuit curve In geometry, a curve of pursuit is a curve constructed by analogy to having a point or points representing pursuers and pursuees; the curve of pursuit is the curve traced by the pursuers. With the paths of the pursuer and pursuee parameterized ...

File:Loxodrome.png,

Rhumb line In navigation, a rhumb line, rhumb (), or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true north. Introduction The effect of following a rhumb li ...

File:Syntractrix_a%3D0.5_b%3D1.png, Syntractrix File:Tractrix.png,

Tractrix In geometry, a tractrix (; plural: tractrices) is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a pulling point (the ''tractor'') that moves at a right angl ...

File:Cycloids.svg, Trochoid

Spirals

File:Archimedean_spiral_polar.svg,

Archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...

File:Euler_spiral.svg,

Cornu spiral An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). Euler spirals are also commonly referred to as spiros, clothoids, or Cornu spirals. E ...

File:Fermat%27s_spiral.svg,

Fermat's spiral A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance f ...

File:Hyperspiral.svg,

Hyperbolic spiral A hyperbolic spiral is a plane curve, which can be described in polar coordinates by the equation :r=\frac of a hyperbola. Because it can be generated by a circle inversion of an Archimedean spiral, it is called Reciprocal spiral, too.. Pierre ...

File:Lituus.svg, Lituus File:Logarithmic_Spiral_Pylab.svg, Logarithmic spiral

Piecewise constructions

File:Maurer_roses.svg, Maurer rose File:ReuleauxTriangle.svg,

Reuleaux triangle A Reuleaux triangle is a curved triangle with constant width, the simplest and best known curve of constant width other than the circle. It is formed from the intersection of three circular disks, each having its center on the boundary of the ...

Fractal curves

File:Blancmange-function.svg,

Blancmange curve In mathematics, the blancmange curve is a self-affine curve constructible by midpoint subdivision. It is also known as the Takagi curve, after Teiji Takagi who described it in 1901, or as the Takagi–Landsberg curve, a generalization of the curv ...

File:Cesaro-0.5.png, De Rham curve File:Fractal_dragon_curve.jpg,

Dragon curve A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems. The dragon curve is probably most commonly thought of as the shape that is generated from repe ...

File:KochFlake.svg, Koch curve File:Levy_C_construction.png,

Lévy C curve In mathematics, the Lévy C curve is a self-similar fractal curve that was first described and whose differentiability properties were analysed by Ernesto Cesàro in 1906 and Georg Faber in 1910, but now bears the name of French mathematician Pa ...

File:Peanocurve.svg,

Peano curve In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injective. ...

File:Sierpinski-Curve-3.png,

Sierpiński curve Sierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit n \to \infty completely fill the unit square: thus their limit curve, also called the Sierpińs ...

External links

Visual Dictionary of Special Plane Curves
{{Curves * Image galleries