What is a Curve ?

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 everyday 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.

A closed curve is a curve that forms a path whose starting point is also its ending point—that is, a path from any of its points to the same point.

Closely related meanings include the graph of a function (for example, Phillips curve) and a two-dimensional graph

Interest in curves began long before they were the subject of mathematical study.

This can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times

Curves, or at least their graphical representations, are simple to create, for example by a stick in the sand on a beach.

Historically, the term "line" was used in place of the more modern term "curve". Hence the phrases "straight line" and "right line" were used to distinguish what are today called lines from "curved lines". For example, in Book I of Euclid's Elements, a line is defined as a "breadthless length" (Def. 2), while a straight line is defined as "a line that lies evenly with the points on itself" (Def. 4). Euclid's idea of a line is perhaps clarified by the statement "The extremities of a line are points," (Def. 3)

The Greek geometers had studied many other kinds of curves

One reason was their interest in solving geometrical problems that could not be solved using standard compass and straightedge construction. These curves include:

The conic sections, deeply studied by Apollonius of Perga
The cissoid of Diocles, studied by Diocles and used as a method to double the cube.

The conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle

The Archimedean spiral, studied by Archimedes as a method to trisect an angle and square the circle

The spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius.

A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century.

This enabled a curve to be described using an equation rather than an elaborate geometrical construction.

This not only allowed new curves to be defined and studied, but it enabled a formal distinction to be made between curves that can be defined using algebraic equations, algebraic curves, and those that cannot, transcendental curves.

Previously, curves had been described as "geometrical" or "mechanical" according to how they were, or supposedly could be, generated.

Conic sections were applied in astronomy by Kepler. Newton also worked on an early example in the calculus of variations

Solutions to variational problems, such as the brachistochrone and tautochrone questions, introduced properties of curves in new ways (in this case, the cycloid).

The catenary gets its name as the solution to the problem of a hanging chain, the sort of question that became routinely accessible by means of differential calculus.

In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the cubic curves, in the general description of the real points into 'ovals'

The statement of Bézout's theorem showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions.

Since the nineteenth century there has not been a separate theory of curves, but rather the appearance of curves as the one-dimensional aspect of projective geometry, and differential geometry; and later topology, when for example the Jordan curve theorem was understood to lie quite deep, as well as being required in complex analysis.

The era of the space-filling curves finally provoked the modern definitions of curve.