# 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 $\gamma :{\mathbb{S}}^{1}\to X$ 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:

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

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^{} $\overrightarrow{\gamma}:I\to {\mathbb{R}}^{n}$, that is an
$n$-tuple of functions

$$\overrightarrow{\gamma}(t)=\left(\begin{array}{c}\hfill {\gamma}_{1}(t)\hfill \\ \hfill \mathrm{\vdots}\hfill \\ \hfill {\gamma}_{n}(t)\hfill \end{array}\right),$$ |

where ${\gamma}_{i}:I\to \mathbb{R}$, $i=1,\mathrm{\dots},n$ 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 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

$$ |

for some real $\u03f5>0$.

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

$${\gamma}_{i}:{I}_{i}\to X,i=1,2$$ |

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 |