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The original formulation of the Schoenflies problem states that not only does every simple closed curve in the plane separate the plane into two regions, one (the "inside") bounded and the other (the "outside") unbounded; but also that these two regions are homeomorphic to the inside and outside of a standard circle in the plane.
Figure 1: Zindler curve. Any of the chords of equal length cuts the curve and the enclosed area into halves. Figure 2: Examples of Zindler curves with a = 8 (blue), a = 16 (green) and a = 24 (red). A Zindler curve is a simple closed plane curve with the defining property that: (L) All chords which cut the curve length into halves have the same ...
A plane curve is the image of any continuous function from an interval to the Euclidean plane.Intuitively, it is a set of points that could be traced out by a moving point. More specifically, smooth curves generally at least require that the function from the interval to the plane be continuously differentiable, and in some contexts are defined to require higher derivative
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.
A smooth plane curve is a curve in a real Euclidean plane and is a one-dimensional smooth manifold.This means that a smooth plane curve is a plane curve which "locally looks like a line", in the sense that near every point, it may be mapped to a line by a smooth function.
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? This is true if the curve is convex or piecewise smooth and in other special cases. The problem was proposed by Otto Toeplitz in 1911. [1]
Two Dimensional Curves; Visual Dictionary of Special Plane Curves; Curves and Surfaces Index (Harvey Mudd College) National Curve Bank; An elementary treatise on cubic and quartic curves by Alfred Barnard Basset (1901) online at Google Books
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 ]