Curve

In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. Simple examples are the circle or the straight line. A large number of other curves have been studied in geometry.

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{R}). Then a curve !,gamma is a continuous mapping ,!gamma : I ightarrow 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) ightarrow x = y. If I is a closed bounded interval ,!, we also allow the possibility ,!gamma(a) = gamma(b) (this convention makes it possible to talk about closed simple curve).

Related Topics:
Mathematics - Interval - Real number - Non-empty - Connected - Subset - Continuous - Mapping - Topological space - Injective

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If gamma(x)=gamma(y) for some x e y (other than the extremities of I), then gamma(x) is called a double (or: multiple) point of the curve.

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A curve !,gamma is said to be closed or a loop if ,!I = 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.

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A plane curve is a curve for which X is the mathematical 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).

Related Topics:
Mathematical plane - Projective plane - Euclidean space

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This definition of curve captures our intuitive notion of a curve as a connected, continuous geometric figure that is "like" a line, although it also includes figures that can be hardly called curves in common usage. For example, the image of a curve can cover a square in the plane (Peano 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 yet another weird example.

Related Topics:
Square - Peano curve - Hausdorff dimension - Koch snowflake - Positive - Lebesgue measure - Dragon curve

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