curve

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

⊂

ℝ

be an

interval

(

http://planetmath.org/Interval

) of the

real line

. A parameterized
curve is a

continuous mapping

γ

:

I

→

X

taking values in a

topological space

X

. We say that

γ

is a

simple curve

if it has no self-intersections, that is if the

mapping

γ

is

injective

.

We say that

γ

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;

γ

⁢

(

a

)

=

γ

⁢

(

b

)

.

Equivalently, a loop may be defined to be a
continuous mapping

γ

:

𝕊

1

→

X

whose domain

𝕊

1

is the

unit circle

. A simple closed curve is often
called a

Jordan curve

.

If

X

=

ℝ

2

then

γ

is called a

plane curve

or

planar curve

.

A smooth closed curve

γ

in

ℝ

n

is

locally

if the local multiplicity of

intersection

of

γ

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

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

ℝ

2

⁢

n

is the normalized
generalized ellipse:

(

sin

⁡

t

,

cos

⁡

t

,

sin

⁡

2

⁢

t

2

,

cos

⁡

2

⁢

t

2

,

…

,

sin

⁡

n

⁢

t

n

,

cos

⁡

n

⁢

t

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

γ

:

I

→

X

be a differentiable curve. For
every

t

∈

I

can regard the

derivative

(

http://planetmath.org/RelatedRates

),

γ

˙

⁢

(

t

)

, as the

velocity

(

http://planetmath.org/RelatedRates

) of a
moving particle, at time

t

. The velocity

γ

˙

⁢

(

t

)

is a

tangent vector

(

http://planetmath.org/TangentSpace

), which belongs to

T

γ

⁢

(

t

)

⁢

X

, the

tangent space

of the

manifold

X

at the point

γ

⁢

(

t

)

. We say that a differentiable curve

γ

⁢

(

t

)

is

regular

, if its velocity,

γ

˙

⁢

(

t

)

, is non-vanishing
for all

t

∈

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 a

vector-valued function

γ

→

:

I

→

ℝ

n

, that is an

n

-tuple of functions

γ

→

⁢

(

t

)

=

(

γ

1

⁢

(

t

)

⋮

γ

n

⁢

(

t

)

)

,

where

γ

i

:

I

→

ℝ

,

i

=

1

,

…

,

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

∈

C

there exists an open

neighbourhood

U

⊂

X

of

p

and a chart

α

:

U

→

ℝ

n

such that

α

⁢

(

C

∩

U

)

=

{

(

t

,

0

,

…

,

0

)

∈

ℝ

n

:

-

ϵ

<

t

<

ϵ

}

for some real

ϵ

>

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

⊂

ℝ

be intervals. A reparameterization is a

continuously
differentiable function

s

:

I

1

→

I

2

whose derivative is never vanishing. Thus,

s

is
either

monotone increasing

, or

monotone decreasing

.
Two regular, parameterized curves

γ

i

:

I

i

→

X

,

i

=

1

,

2

are said to be related by a reparameterization if there exists a
reparameterization

s

:

I

1

→

I

2

such that

γ

1

=

γ

2

∘

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

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