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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.
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.
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 ...
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 ]
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
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The term oval when used to describe curves in geometry is not well-defined, except in the context of projective geometry. Many distinct curves are commonly called ovals or are said to have an "oval shape". Generally, to be called an oval, a plane curve should resemble the outline of an egg or an ellipse. In particular, these are common traits ...
An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0.This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.