Closed curve

A curve whose endpoints coincide, forming a complete loop. It may be simple, with no self-intersections, or self-intersecting.

Curve: A continuous path in a space, often represented by a parametrized map from an interval. A closed curve is a curve whose parametrization returns to its starting point.
Closed path: A continuous path whose starting and ending points are the same. This path-based definition captures closure independently of a particular drawing.
Contour integration: Integration of complex-valued functions along oriented curves in the complex plane. Closed contours let integral theorems relate boundary values to singularities inside.
Open curve: A curve whose endpoints are distinct. It differs from a closed curve precisely because its traversal does not return to its start.
Parametric curve: A curve specified by coordinate functions of a parameter. Matching the parameter interval’s endpoint values makes a parametrized curve closed.
Arc length: The distance measured along a curve. A regular closed curve has a total length accumulated over one complete traversal.
Green's function: A function used to solve linear differential equations by representing a system’s response to a point source. Closed-curve boundaries often specify the domains and boundary conditions in these methods.
Knot theory: The mathematical study of embeddings of circles in three-dimensional space. Knot theory studies closed curves in space up to continuous deformation without cutting.
Simple curve: A curve that does not intersect itself, apart from coincident endpoints when it is closed. Simplicity is an extra condition; closed curves can also cross themselves.
Winding number: An integer measuring how many times a closed curve winds around a point. It records a loop’s turning around points not on the curve.

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