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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.
The Jordan curve theorem states that every simple closed curve has a well-defined "inside" and "outside";
A plane simple closed curve is also called a Jordan curve. It is also defined as a non-self-intersecting continuous loop in the plane. [ 9 ] The Jordan curve theorem states that the set complement in a plane of a Jordan curve consists of two connected components (that is the curve divides the plane in two non-intersecting regions that are both ...
Toggle Mathematics (Geometry) subsection. 1.1 Algebraic curves. ... This is a list of Wikipedia articles about curves used in different fields: mathematics ...
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.
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.
Example: The black dashed curve goes through all corners of several blue squares. The inscribed square problem, also known as the square peg problem or the Toeplitz' conjecture, is an unsolved question in geometry: Does every plane simple closed curve contain all four vertices of some square?
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 ]