# curve

## Summary.

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 geometry, the term curve is used to describe a 1-dimensional variety 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 geometry, one would refer to as a surface. In particular, a Riemann surface can be regarded as as complex curve.

## Kinematic definition

Let $I\subset\mathbb{R}$ be an interval (http://planetmath.org/Interval) of the real line. A parameterized curve is a continuous mapping $\gamma:I\to X$ taking values in a topological space $X$. We say that $\gamma$ is a simple curve if it has no self-intersections, that is if the mapping $\gamma$ is injective.

We say that $\gamma$ is a closed curve, or a loop (http://planetmath.org/loop) whenever $I=[a,b]$ is a closed interval, and the endpoints are mapped to the same value; $\gamma(a)=\gamma(b).$ Equivalently, a loop may be defined to be a continuous mapping $\gamma\colon\mathbb{S}^{1}\to X$ whose domain $\mathbb{S}^{1}$ is the unit circle. A simple closed curve is often called a Jordan curve.

If $X=\mathbb{R}^{2}$ then $\gamma$ is called a plane curve or planar curve.

A smooth closed curve $\gamma$ in $\mathbb{R}^{n}$ is locally if the local multiplicity of intersection of $\gamma$ with each hyperplane 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 $\mathbb{R}^{n}$ 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 $\mathbb{R}^{2n}$ is the normalized generalized ellipse:

 $(\sin t,\cos t,\frac{\sin 2t}{2},\frac{\cos 2t}{2},\ldots,\frac{\sin nt}{n},% \frac{\cos nt}{n}).$

In odd dimension there are no closed convex curves.

In many instances the ambient space $X$ is a differential manifold, in which case we can speak of differentiable curves. Let $I$ be an open interval, and let $\gamma:I\to X$ be a differentiable curve. For every $t\in I$ can regard the derivative (http://planetmath.org/RelatedRates), $\dot{\gamma}(t)$, as the velocity (http://planetmath.org/RelatedRates) of a moving particle, at time $t$. The velocity $\dot{\gamma}(t)$ is a tangent vector (http://planetmath.org/TangentSpace), which belongs to $T_{\gamma(t)}X$, the tangent space of the manifold $X$ at the point $\gamma(t)$. We say that a differentiable curve $\gamma(t)$ is regular, if its velocity, $\dot{\gamma}(t)$, is non-vanishing for all $t\in I$.

It is also quite common to consider curves that take values in $\mathbb{R}^{n}$. In this case, a parameterized curve can be regarded as a vector-valued function $\vec{\gamma}:I\to\mathbb{R}^{n}$, that is an $n$-tuple of functions

 $\vec{\gamma}(t)=\begin{pmatrix}\gamma_{1}(t)\\ \vdots\\ \gamma_{n}(t)\end{pmatrix},$

where $\gamma_{i}:I\to\mathbb{R}$, $i=1,\ldots,n$ are scalar-valued functions.

## Geometric definition.

A (non-singular) curve $C$, equivalently, an arc, is a connected, 1-dimensional submanifold of a differential manifold $X$. This means that for every point $p\in C$ there exists an open neighbourhood $U\subset X$ of $p$ and a chart $\alpha:U\to\mathbb{R}^{n}$ such that

 $\alpha(C\cap U)=\{(t,0,\ldots,0)\in\mathbb{R}^{n}:-\epsilon

for some real $\epsilon>0$.

An alternative, but equivalent definition, describes an arc as the image of a regular parameterized curve. To accomplish this, we need to define the notion of reparameterization. Let $I_{1},I_{2}\subset\mathbb{R}$ be intervals. A reparameterization is a continuously differentiable function

 $s:I_{1}\to I_{2}$

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

 $\gamma_{i}:I_{i}\to X,\quad i=1,2$

are said to be related by a reparameterization if there exists a reparameterization $s:I_{1}\to I_{2}$ such that

 $\gamma_{1}=\gamma_{2}\circ s.$

The inverse of a reparameterization function is also a reparameterization. Likewise, the composition of two parameterizations is again a reparameterization. Thus the reparameterization relation between curves, is in fact an equivalence relation. An arc can now be defined as an equivalence class of regular, simple curves related by reparameterizations. In order to exclude pathological embeddings 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