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A closed curve is thus the image of a continuous mapping of a circle. A non-closed curve may also be called an open curve. If the domain of a topological curve is a closed and bounded interval = [,], the curve is called a path, also known as topological arc (or just arc).
A Jordan curve or a simple closed curve in the plane R 2 is the image C of an injective continuous map of a circle into the plane, φ: S 1 → R 2. A Jordan arc in the plane is the image of an injective continuous map of a closed and bounded interval [a, b] into the plane. It is a plane curve that is not necessarily smooth nor algebraic.
By the Jordan curve theorem, a simple closed curve divides the plane into interior and exterior regions, and another equivalent definition of a closed convex curve is that it is a simple closed curve whose union with its interior is a convex set. [9] [17] Examples of open and unbounded convex curves include the graphs of convex functions.
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that the curve travels counterclockwise around the point, i.e., the curve's number of turns. For certain open plane curves, the number of turns may be a non-integer.
This definition relies on the fact that every simple closed curve admits a well-defined interior, which follows from the Jordan curve theorem. The inner loop of a beltway road in a country where people drive on the right side of the road is an example of a negatively oriented curve.
This definition generalizes to any subset of a metric space with metric : is an interior point of if there exists a real number >, such that is in whenever the distance (,) <. This definition generalizes to topological spaces by replacing "open ball" with " open set ".
In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width or an orbiform , the name given to these shapes by Leonhard Euler . [ 1 ]
When L is a closed curve (initial and final points coincide), the line integral is often denoted (), sometimes referred to in engineering as a cyclic integral. To establish a complete analogy with the line integral of a vector field, one must go back to the definition of differentiability in multivariable calculus.