In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but which is not required to be straight. This entails that a line is a special case of curve, namely a curve with null curvature. Often curves in two-dimensional (plane curves) or three-dimensional (space curves) Euclidean space are of interest.
Various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However many of these meanings are special instances of the definition which follows. A curve is a topological space which is locally homeomorphic to a line. In every day language, this means that a curve is a set of points which, near each of its points, looks like a line, up to a deformation. A simple example of a curve is the parabola, shown to the right. A large number of other curves have been studied in multiple mathematical fields.
The term curve has several meanings in non-mathematical language as well. For example, it can be almost synonymous with mathematical function (as in learning curve), or graph of a function (as in Phillips curve).
An arc or segment of a curve is a part of a curve that is bounded by two distinct end points and contains every point on the curve between its end points. Depending on how the arc is defined, either of the two end points may or may not be part of it. When the arc is straight, it is typically called a line segment.
Other articles related to "curve, curves":
... Curve Lake is the name of two Ojibwe Indian reserves 14 km north of Peterborough, Ontario ... They serve as the landbase for the Curve Lake First Nation ... Officially they are known as Curve Lake First Nation No ...
... A parallel of a curve is the envelope of a family of congruent circles centered on the curve ... It can also be defined as a curve whose points are at a fixed normal distance of a given curve ... It is sometimes called the offset curve but the term "offset" often refers also to translation ...
... Elliptic curve Diffie–Hellman (ECDH) is an anonymous key agreement protocol that allows two parties, each having an elliptic curve public-private key pair, to establish a shared secret over an ... Diffie–Hellman protocol using elliptic curve cryptography ...
... A curve in a two dimensional space is best represented by the parametric equations like x(c) and y(c) ... The family of curves can be represented in the form where c is the parameter ... The envelope of a family of curves g(x,y,c) = 0 is a curve such that at each point on the curve there is some member of the family that touches that ...
... Algebraic curves are the curves considered in algebraic geometry ... A plane algebraic curve is the locus of the points of coordinates x, y such that f(x, y) = 0, where f is a polynomial in two variables defined over some field F ... If C is a curve defined by a polynomial f with coefficients in F, the curve is said defined over F ...
Famous quotes containing the word curve:
“And out again I curve and flow
To join the brimming river,
For men may come and men may go,
But I go on forever.”
—Alfred Tennyson (18091892)
“In philosophical inquiry, the human spirit, imitating the movement of the stars, must follow a curve which brings it back to its point of departure. To conclude is to close a circle.”
—Charles Baudelaire (18211867)
“I have been photographing our toilet, that glossy enameled receptacle of extraordinary beauty.... Here was every sensuous curve of the human figure divine but minus the imperfections. Never did the Greeks reach a more significant consummation to their culture, and it somehow reminded me, in the glory of its chaste convulsions and in its swelling, sweeping, forward movement of finely progressing contours, of the Victory of Samothrace.”
—Edward Weston (18861958)