Jordan Curve

Jordan curve [zhȯr′dän‚kərv]

(mathematics)

A simple closed curve in the plane, that is, a curve that is closed, connected, and does not cross itself.

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Jordan Curve 

the geometric locus of points M(x, y) of a plane, the coordinates of which satisfy the equations x = ϕ(t) and y = ψ(t) where ϕ and ψ are continuous functions of the argument t on some segment [a, b]. Stated differently, the Jordan curve is a continuous image of the segment [a, b]. This definition is one of several possible definitions of a continuous curve that are mathematically rigorous. However, the Jordan curve may have very little in common with the conventional idea of a curve; for instance, the Jordan curve may pass through all points of a certain square.

If the points M(x, y) of a Jordan curve, corresponding to different values of t, are different from one another, such a Jordan curve is called a simple arc. In other words, a simple arc is a Jordan curve without multiple points. A simple arc is a homeomorphic image of a segment. If the points of a Jordan curve, corresponding to t = a and t = b, coincide and all other points are different from one another and different from M[ϕ(a), ψ(a)], then the Jordan curve is called a simple closed curve. Such a Jordan curve is a homeomorphic image of a circle.

The French mathematician Camille Jordan, after whom the curve is named, showed in 1882 that any closed Jordan curve without multiple points divides a plane into two regions, one of which is the interior with respect to this curve and the other is the exterior. This proposition is called the Jordan curve theorem.

S. B. STECHKIN