# Dictionary Definition

curve

### Noun

1 the trace of a point whose direction of motion
changes [syn: curved
shape] [ant: straight
line]

2 a line on a graph representing data

3 a baseball thrown with spin so that its path
curves as it approach the batter [syn: curve ball,
breaking
ball, bender]

4 the property possessed by the curving of a line
or surface [syn: curvature]

5 curved segment (of a road or river or railroad
track etc.) [syn: bend]

### Verb

1 turn sharply; change direction abruptly; "The
car cut to the left at the intersection"; "The motorbike veered to
the right" [syn: swerve,
sheer, trend, veer, slue, slew, cut]

2 extend in curves and turns; "The road winds
around the lake" [syn: wind]

4 bend or cause to bend; "He crooked his index
finger"; "the road curved sharply" [syn: crook]

# User Contributed Dictionary

## English

### Etymology

From etyl la curvus### Adjective

curve- Bent without angles; crooked; curved.
- a curve line
- a curve surface

### Noun

- A gentle bend, such as in a road.
- A simple figure containing no straight portions and no angles; a curved line.
- A continuous map from a one-dimensional space to a multidimensional space.
- A one-dimensional figure of non-zero length; the graph of a continuous map from a one-dimensional space.
- An algebraic curve; a polynomial relation of the planar coordinates.
- A one-dimensional continuum.
- (informal, usually in plural curves) The attractive shape of a woman's body.

#### Derived terms

- algebraic curve
- Bézier curve
- closed curve
- cosine curve
- curvy
- dragon curve
- elliptic curve
- Lissajous curve
- Jordan curve
- nonsimple curve
- open curve
- pedal curve
- plane curve
- pursuit curve
- simple curve
- sine curve
- space curve
- spherical curve

#### Translations

gentle bend

geometry: one-dimensional figure

algebraic curve

- See algebraic curve

informal: usually in plural: attractive features
of a woman

- Hebrew: חיטובים (khit'uvym) m|p

informal: attractive shape of a woman's body

### Verb

- To bend; to crook.
- to curve a line
- to curve a pipe

- To cause to swerve from a straight course.
- to curve a ball in pitching it

- To bend or turn gradually from a given direction.
- the road curves to the right

#### Translations

bend, crook

cause to swerve from a straight course

bend or turn gradually from a given direction

- ttbc Indonesian: kurva
- ttbc Irish: cuar

## Italian

### Noun

curve- Plural of curva

### Adjective

curve- feminine plural of curvo

## Romanian

### Pronunciation

### Noun

curve f|p- Plural of curvă whores

# 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

aberrancy, aberration, arc, arch, artful dodge, artifice, bag of tricks, bear
off, bend, bend back,
bias, blind, bluff, bosey, bow, bowl, branch off, branching off,
cast, catacaustic, catch, catenary, caustic, change of pace, change
the bearing, change-up, chicanery, chouse, chuck, chunk, circle, circuit, circuitousness, circumference, coil, compass, conchoid, corner, crook, curl, curvation, curvature, curve-ball,
declination,
decurve, deflect, depart from, departure, design, detour, deviance, deviancy, deviate, deviation, device, deviousness, diacaustic, diffract, diffuse, digress, digression, dirty deal, dirty
trick, discursion,
disperse, distort, divagate, divagation, divaricate, divarication, diverge, divergence, diversion, divert, dodge, dogleg, dome, double, downcurve, drift, drifting, ellipse, embow, errantry, excursion, excursus, exorbitation, fast deal,
fastball, feint, festoon, fetch, ficelle, flex, fling, flip, forward pass, gambit, gimmick, googly, hairpin, heave, heel, hocus-pocus, hook, hump, hunch, hurl, hyperbola, incurvate, incurvation, incurvature, incurve, indirection, inflect, inflection, joker, juggle, knuckleball, lateral, lateral pass, lituus, lob, loop, obliquity, outcurve, parabola, pass, peg, pererration, pitch, ploy, pull, put, rambling, recurve, reflect, reflex, refract, retroflex, rondure, round, ruse, sag, scatter, scheme, screwball, scurvy trick,
serve, service, sheer, shift, shifting, shifting course,
shifting path, shot-put, shy, sinker, sinus, skew, slant, sleight, sleight of hand,
sleight-of-hand trick, slider, sling, spiral, spitball, spitter, stratagem, straying, subterfuge, swag, sweep, swerve, swerving, swinging, tack, throw, toss, tracery, trend, trick, turn, turn aside, turning, twist, upcurve, variation, vary, vault, veer, wandering, warp, wile, wind, yaw, zigzag