Curve

Intuitively, a curve may be thought of as the trace left by a moving point.

This is the definition that appeared more than 2000 years ago in Euclid's Elements: "The [curved] line[a] is […] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width.

A plane algebraic curve is the zero set of a polynomial in two indeterminates.

When complex zeros are considered, one has a complex algebraic curve, which, from the topological point of view, is not a curve, but a surface, and is often called a Riemann surface.

In particular, algebraic curves over a finite field are widely used in modern cryptography.

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.

[2] Curves, or at least their graphical representations, are simple to create, for example with a stick on the sand on a beach.

For example, in Book I of Euclid's Elements, a line is defined as a "breadthless length" (Def.

For example:[4] The Greek geometers had studied many other kinds of curves.

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.

A curve is simple if it is the image of an interval or a circle by an injective continuous function.

[10] Fractal curves can have properties that are strange for the common sense.

For example, a fractal curve can have a Hausdorff dimension bigger than one (see Koch snowflake) and even a positive area.

In Euclidean geometry, an arc (symbol: ⌒) is a connected subset of a differentiable curve.

is which can be thought of intuitively as using the Pythagorean theorem at the infinitesimal scale continuously over the full length of the curve.

The needs of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions.

This general idea is enough to cover many of the applications of curves in mathematics.

On the other hand, it is useful to be more general, in that (for example) it is possible to define the tangent vectors to

is an analytic manifold (i.e. infinitely differentiable and charts are expressible as power series), and

A plane algebraic curve is the set 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. One says that the curve is defined over F. Algebraic geometry normally considers not only points with coordinates in F but all the points with coordinates in an algebraically closed field K. If C is a curve defined by a polynomial f with coefficients in F, the curve is said to be defined over F. In the case of a curve defined over the real numbers, one normally considers points with complex coordinates.

In particular, the nonsingular complex projective algebraic curves are called Riemann surfaces.

The points of a curve C with coordinates in a field G are said to be rational over G and can be denoted C(G).

By eliminating variables (by any tool of elimination theory), an algebraic curve may be projected onto a plane algebraic curve, which however may introduce new singularities such as cusps or double points.

An example is the Fermat curve un + vn = wn, which has an affine form xn + yn = 1.

A similar process of homogenization may be defined for curves in higher dimensional spaces.