The term curve is associated with two closely related notions. The first notion is kinematic: a parameterized curve is a function of one real variable taking values in some ambient geometric setting. This variable can be interpreted as time, in which case the function describes the evolution of a moving particle. The second notion is geometric; in this sense a curve is an arc, a 1-dimensional subset of an ambient space. The two notions are related: the image of a parameterized curve describes the trajectory of a moving particle. Conversely, a given arc admits multiple parameterizations. A trajectory can be traversed by moving particles at different speeds.

In algebraic geometryMathworldPlanetmathPlanetmath, the term curve is used to describe a 1-dimensional varietyMathworldPlanetmathPlanetmath relative to the complex numbers or some other ground field. This can be potentially confusing, because a curve over the complex numbers refers to an object which, in conventional geometryMathworldPlanetmath, one would refer to as a surfaceMathworldPlanetmath. In particular, a Riemann surface can be regarded as as complex curve.

Kinematic definition

Let I be an intervalMathworldPlanetmathPlanetmath ( of the real line. A parameterized curve is a continuous mapping γ:IX taking values in a topological spaceMathworldPlanetmath X. We say that γ is a simple curve if it has no self-intersections, that is if the mapping γ is injectivePlanetmathPlanetmath.

We say that γ is a closed curve, or a loop ( whenever I=[a,b] is a closed interval, and the endpoints are mapped to the same value; γ(a)=γ(b). Equivalently, a loop may be defined to be a continuous mapping γ:𝕊1X whose domain 𝕊1 is the unit circle. A simple closed curve is often called a Jordan curve.

If X=2 then γ is called a plane curve or planar curve.

A smooth closed curve γ in n is locally if the local multiplicity of intersectionMathworldPlanetmath of γ with each hyperplaneMathworldPlanetmathPlanetmath at of each of the intersection points does not exceed n. The global multiplicity is the sum of the local multiplicities. A simple smooth curve in n is called (or globally ) if the global multiplicity of its intersection with any affine hyperplane is less than or equal to n. An example of a closed convex curve in 2n is the normalized generalized ellipse:


In odd dimensionMathworldPlanetmathPlanetmathPlanetmath there are no closed convex curves.

In many instances the ambient space X is a differential manifold, in which case we can speak of differentiableMathworldPlanetmathPlanetmath curves. Let I be an open interval, and let γ:IX be a differentiable curve. For every tI can regard the derivativePlanetmathPlanetmath (, γ˙(t), as the velocity ( of a moving particle, at time t. The velocity γ˙(t) is a tangent vectorMathworldPlanetmath (, which belongs to Tγ(t)X, the tangent spaceMathworldPlanetmath of the manifold X at the point γ(t). We say that a differentiable curve γ(t) is regularPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, if its velocity, γ˙(t), is non-vanishing for all tI.

It is also quite common to consider curves that take values in n. In this case, a parameterized curve can be regarded as a vector-valued functionPlanetmathPlanetmath γ:In, that is an n-tuple of functions


where γi:I, i=1,,n are scalar-valued functions.

Geometric definition.

A (non-singularPlanetmathPlanetmath) curve C, equivalently, an arc, is a connected, 1-dimensional submanifoldMathworldPlanetmath of a differential manifold X. This means that for every point pC there exists an open neighbourhood UX of p and a chart α:Un such that


for some real ϵ>0.

An alternative, but equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath definition, describes an arc as the image of a regular parameterized curve. To accomplish this, we need to define the notion of reparameterization. Let I1,I2 be intervals. A reparameterization is a continuously differentiable function


whose derivative is never vanishing. Thus, s is either monotone increasing, or monotone decreasing. Two regular, parameterized curves


are said to be related by a reparameterization if there exists a reparameterization s:I1I2 such that


The inversePlanetmathPlanetmathPlanetmathPlanetmath of a reparameterization function is also a reparameterization. Likewise, the composition of two parameterizations is again a reparameterization. Thus the reparameterization relationMathworldPlanetmath between curves, is in fact an equivalence relation. An arc can now be defined as an equivalence classMathworldPlanetmath of regular, simple curves related by reparameterizations. In order to exclude pathological embeddingsMathworldPlanetmathPlanetmath with wild endpoints we also impose the condition that the arc, as a subset of X, be homeomorphic to an open interval.

Title curve
Canonical name Curve
Date of creation 2013-03-22 12:54:17
Last modified on 2013-03-22 12:54:17
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 28
Author rmilson (146)
Entry type Definition
Classification msc 53B25
Classification msc 14H50
Classification msc 14F35
Classification msc 51N05
Synonym parametrized curve
Synonym parameterized curve
Synonym path
Synonym trajectory
Related topic FundamentalGroup
Related topic TangentSpace
Related topic RealTree
Defines closed curve
Defines Jordan curve
Defines regular curve
Defines simple closed curve
Defines simple curve
Defines plane curve
Defines planar curve
Defines convex curve
Defines locally convex curve
Defines local multiplicity
Defines globally convex
Defines global multiplicity