Curve

History

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). Later commentators further classified lines according to various schemes. For example:

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:

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.

From the nineteenth century there is not a separate curve theory, 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.

Definition

In general, a curve is defined through a continuous function γ : I → X from an interval I of the real numbers into a topological space X. Depending on the context, it is either γ or its image γ ( I ) which is called a curve.

In general topology, when non-differentiable functions are considered, it is the map γ , which is called a curve, because its image may look very differently from what is commonly called a curve. For example, the image of the Peano curve completely fills the square. On the other hand, when one considers curves defined by a differentiable function (or, at least, a piecewise differentiable function), this is commonly the image of the function which is called a curve.

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 of plane, space and skew curves apply also to real algebraic curves, although the above definition of a curve does not applies (a real algebraic curve may be disconnected).

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.

Differentiable curve

Roughly speaking a differentiable curve is a curve that is defined as being locally the image of an injective differentiable function γ : I → X from an interval I of the real numbers into a differentiable manifold X, often R n .

More precisely, a differentiable curve is a subset C of X where every point of C has a neighborhood U such that C ∩ U is diffeomorphic to an interval of the real numbers. In other words, a differentiable curve differentiable manifold is of dimension one.

Length of a curve

If X = R n is the n -dimensional Euclidean space, and if γ : [ a , b ] → R n is an injective and continuously differentiable function, then the length of γ is defined as the quantity

The length of a curve is independent of the parametrization γ .

In particular, the length s of the graph of a continuously differentiable function y = f ( x ) defined on a closed interval [ a , b ] is

More generally, if X is a metric space with metric d , then we can define the length of a curve γ : [ a , b ] → X by

where the supremum is taken over all n ∈ N and all partitions t 0 < t 1 < … < t n of [ a , b ] .

A rectifiable curve is a curve with finite length. A curve γ : [ a , b ] → X is called natural (or unit-speed or parametrized by arc length) if for any t 1 , t 2 ∈ [ a , b ] such that t 1 ≤ t 2 , we have

If γ : [ a , b ] → X is a Lipschitz-continuous function, then it is automatically rectifiable. Moreover, in this case, one can define the speed (or metric derivative) of γ at t ∈ [ a , b ] as

and then show that

Differential geometry

While the first examples of curves that are met are mostly plane curves (that is, in everyday words, curved lines in two-dimensional space), there are obvious examples such as the helix which exist naturally in three dimensions. The needs of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions. In general relativity, a world line is a curve in spacetime.

If X is a differentiable manifold, then we can define the notion of differentiable curve in X . This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can take X to be Euclidean space. On the other hand, it is useful to be more general, in that (for example) it is possible to define the tangent vectors to X by means of this notion of curve.

If X is a smooth manifold, a smooth curve in X is a smooth map

This is a basic notion. There are less and more restricted ideas, too. If X is a C k manifold (i.e., a manifold whose charts are k times continuously differentiable), then a C k curve in X is such a curve which is only assumed to be C k (i.e. k times continuously differentiable). If X is an analytic manifold (i.e. infinitely differentiable and charts are expressible as power series), and γ is an analytic map, then γ is said to be an analytic curve.

A differentiable curve is said to be regular if its derivative never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.) Two C k differentiable curves

are said to be equivalent if there is a bijective C k map

such that the inverse map

is also C k , and

for all t . The map γ 2 is called a reparametrisation of γ 1 ; and this makes an equivalence relation on the set of all C k differentiable curves in X . A C k arc is an equivalence class of C k curves under the relation of reparametrisation.

Algebraic curve

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. Algebraic geometry normally looks not only on points with coordinates in F but on 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 defined over F. The points of the curve C with coordinates in a field G are said rational over G and can be denoted C(G)). When G is the field of the rational numbers, one simply talks of rational points. For example, Fermat's Last Theorem may be restated as: For n > 2, every rational point of the Fermat curve of degree n has a zero coordinate.

Algebraic curves can also be space curves, or curves in a space of higher dimension, say n. They are defined as algebraic varieties of dimension one. They may be obtained as the common solutions of at least n–1 polynomial equations in n variables. If n–1 polynomials are sufficient to define a curve in a space of dimension n, the curve is said to be a complete intersection. 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.

A plane curve may also be completed in a curve in the projective plane: if a curve is defined by a polynomial f of total degree d, then w^df(u/w, v/w) simplifies to a homogeneous polynomial g(u, v, w) of degree d. The values of u, v, w such that g(u, v, w) = 0 are the homogeneous coordinates of the points of the completion of the curve in the projective plane and the points of the initial curve are those such w is not zero. 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

Important examples of algebraic curves are the conics, which are nonsingular curves of degree two and genus zero, and elliptic curves, which are nonsingular curves of genus one studied in number theory and which have important applications to cryptography. Because algebraic curves in fields of characteristic zero are most often studied over the complex numbers, algebraic curves in algebraic geometry may be considered as real surfaces. In particular, the non-singular complex projective algebraic curves are called Riemann surfaces.