curve

Summary.

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

In algebraic geometry, the term curve is used to describe a1-dimensional variety relative to the complex numbers or some otherground field. This can be potentially confusing, because a curve overthe complex numbers refers to an object which, in conventionalgeometry, one would refer to as a surface. In particular, a Riemannsurface can be regarded as as complex curve.

Kinematic definition

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

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

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

A smooth closed curve $\gamma$ in ${ℝ}^n$ is locally if the local multiplicity of intersectionof $\gamma$ with each hyperplane at of each of the intersection points does notexceed $n$. The global multiplicity is the sum of the localmultiplicities.A simple smooth curve in ${ℝ}^n$ is called (orglobally ) if the global multiplicityof its intersection with any affine hyperplane is less than or equal to $n$.An example of a closed convex curve in ${ℝ}^{2n}$ is the normalizedgeneralized ellipse:

$$(\sin t,\cos t,\frac{\sin 2t}{2},\frac{\cos 2t}{2},\mathrm{\dots },\frac{\sin nt}{n},\frac{\cos nt}{n}).$$

In odd dimension there are no closed convex curves.

In many instances the ambient space $X$ is a differential manifold, inwhich case we can speak of differentiable curves. Let $I$ be an openinterval, and let $\gamma :I\to X$ be a differentiable curve. Forevery $t\in I$ can regard the derivative (http://planetmath.org/RelatedRates),$\dot{\gamma }(t)$, as the velocity (http://planetmath.org/RelatedRates) of amoving particle, at time $t$. The velocity $\dot{\gamma }(t)$ is atangent 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 $\gamma (t)$ isregular, if its velocity, $\dot{\gamma }(t)$, is non-vanishingfor all $t\in I$.

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

$$\overrightarrow{\gamma }(t)=\left(\begin{array}{c}\hfill {\gamma }_1(t)\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill {\gamma }_n(t)\hfill \end{array}\right),$$

where ${\gamma }_i:I\to ℝ$, $i=1,\mathrm{\dots },n$ 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 meansthat for every point $p\in C$ there exists an open neighbourhood$U\subset X$ of $p$ and a chart $\alpha :U\to {ℝ}^n$ such that

for some real $ϵ>0$.

An alternative, but equivalent definition, describes an arc as theimage of a regular parameterized curve. To accomplish this, we needto define the notion of reparameterization. Let $I_1,I_2\subset ℝ$ be intervals. A reparameterization is a continuouslydifferentiable function

$$s:I_1\to I_2$$

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

$${\gamma }_i:I_i\to X,i=1,2$$

are said to be related by a reparameterization if there exists areparameterization $s:I_1\to I_2$ such that

$${\gamma }_1={\gamma }_2\circ s.$$

The inverse of a reparameterizationfunction is also a reparameterization. Likewise, the composition oftwo parameterizations is again a reparameterization. Thus thereparameterization relation between curves, is in fact anequivalence relation. An arc can now be defined as an equivalenceclass of regular, simple curves related by reparameterizations. Inorder to exclude pathological embeddings with wild endpoints we alsoimpose the condition that the arc, as a subset of $X$, behomeomorphic 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