Plane curves

An oval () is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas of mathematics (projective geometry, technical drawing, etc.), it is given a more precise definition, which may include either one or two axes of symmetry of an ellipse. In common English, the term is used in a broader sense: any shape which reminds one of an egg. The three-dimensional version of an oval is called an ovoid.

spiral with constant width between its turnings; in polar coordinates, distance from the origin is linearly proportional to angle

self-similar growth spiral whose curvature pattern appears frequently in nature

curve connecting two points such that a bead sliding frictionlessly in a uniform gravitational field moves to the other endpoint the fastest

plane algebraic curve

plane curve given by Fresnel integrals such that the curvature increases linearly with curve length

mathematical curve outputted from a specific pair of parametric equations

quartic plane curve defined as the set (or locus) of points in the plane

thumb|500px|Tractrix created by the end of a pole (lying flat on the ground). Its other end is first pushed then dragged by a finger as it spins out to one side.

thumb|400px|right|The lemniscate of Bernoulli and its two foci

mathematical curve

300px|thumb|Examples of superellipses for , .

geometric location

curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane

curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point

sinusoid plotted in polar coordinates

spiral

thumb|200px|right|Squircle centred on the origin () with minor radius :

transcendental curve

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

thumb|upright=1.25|r=\frac{\sin \theta}{\theta}, -20 thumb|upright=1.25|cochleoid (solid) and its polar inverse (dashed) thumb|upright=1.25|A flexible pole is fixed upright at one end and bent over to always form a circular arc. The other end then traces out a Cochleoid.

thumb|300px|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 about the y-axis.

thumb|226px|An epispiral with equation r(θ)=2sec(2θ) The epispiral is a plane curve with polar equation \ r=a \sec{n\theta}. There are n sections if n is odd and 2n if n is even.

quartic plane curve; bicircular quartic curves that are symmetric with respect to the x and y-axes. Spiric sections are included in the family of toric sections and include the family of hippopedes and the family of Cassini ovals

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

type of curve

Type of plane curve

plane quartic curve of the form a²y² – b²x² = x²y²

a non-self-intersecting curve of positive area