# Dictionary Definition

curved adj : not straight; having or marked by a
curve or smoothly rounded bend; "the curved tusks of a walrus";
"his curved lips suggested a smile but his eyes were hard" [syn:
curving] [ant: straight]

# User Contributed Dictionary

## English

### Adjective

- Having a curve or curves.

#### Synonyms

- sense having a curve
- sense having curves curvy, sinuous, tortuous, wavy

#### Translations

having a curve or curves

- French: courbé

### Verb

curved- past of curve

# Extensive Definition

In mathematics, the concept of
a curve tries to capture the intuitive idea of a geometrical
one-dimensional and continuous object. A simple example is the
circle. In everyday use
of the term "curve", a straight line is not curved, but in
mathematical parlance curves include straight lines and line
segments. A large number
of other curves have been studied in geometry.

This article is about the general theory. The
term curve is also used in ways making it almost synonymous with
mathematical
function (as in learning
curve), or graph
of a function (Phillips
curve).

## Definitions

In mathematics, a (topological)
curve is defined as follows. Let I be an interval
of real
numbers (i.e. a non-empty connected
subset of \mathbb). Then
a curve \!\,\gamma is a
continuous mapping
\,\!\gamma : I \rightarrow X, where X is a topological
space. The curve \!\,\gamma is said to be simple if it is
injective, i.e. if for
all x, y in I, we have \,\!\gamma(x) = \gamma(y) \implies x = y. If
I is a closed bounded interval \,\![a, b], we also allow the
possibility \,\!\gamma(a) = \gamma(b) (this convention makes it
possible to talk about closed simple curve). If \gamma(x)=\gamma(y)
for some x\ne y (other than the extremities of I), then \gamma(x)
is called a double (or multiple) point of the curve.

A curve \!\,\gamma is said to be closed or a loop
if \,\!I = [a, b] and if \!\,\gamma(a) = \gamma(b). A closed curve
is thus a continuous mapping of the circle S^1; a simple closed
curve is also called a Jordan curve or a Jordan arc.

A plane curve
is a curve for which X is the Euclidean
plane — these are the examples first encountered
— or in some cases the projective
plane. A space curve is a curve for which X is of three
dimensions, usually Euclidean
space; a skew curve is a space curve which lies in no plane.
These definitions also apply to algebraic
curves (see below). However, in the case of algebraic curves it
is very common not to restrict the curve to having points only
defined over the real numbers.

This definition of curve captures our intuitive
notion of a curve as a connected, continuous geometric figure that
is "like" a line, without thickness and drawn without interruption,
although it also includes figures that can hardly be called curves
in common usage. For example, the image of a curve can cover a
square
in the plane (space-filling
curve). The image of simple plane curve can have Hausdorff
dimension bigger than one (see Koch
snowflake) and even
positive Lebesgue
measure (the last example can be obtained by small variation of
the Peano curve construction). The dragon curve
is another unusual example.

## Conventions and terminology

The distinction between a curve and its image
is important. Two distinct curves may have the same image. For
example, a line segment
can be traced out at different speeds, or a circle can be traversed
a different number of times. Many times, however, we are just
interested in the image of the curve. It is important to pay
attention to context and convention in reading.

Terminology is also not uniform. Often,
topologists use the term "path" for
what we are calling a curve, and "curve" for what we are calling
the image of a curve. The term "curve" is more common in vector
calculus and differential
geometry.

## Lengths of curves

If X is a metric space
with metric d, then we can define the length of a curve \!\,\gamma
: [a, b] \rightarrow X by

- \mbox (\gamma)=\sup \left\

# Synonyms, Antonyms and Related Words

arched,
arciform, bending, bent, billowing, billowy, bowed, curvaceous, curvate, curvated, curve, curvesome, curviform, curvilineal, curvilinear, curving, curvy, embowed, excurved, geosynclinal, incurvate, incurvated, incurved, incurving, labyrinthine, mazy, meandering, recurvate, recurvated, recurved, recurving, round, rounded, serpentine, sinuous, tortuous, twisted, twisting, undulant, wavy