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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.
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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 of ovals: they are differentiable (smooth-looking), [1] simple (not self-intersecting), convex, closed, plane curves;
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.
It is 2 π for convex curves in the plane, and larger for non-convex curves. [1] It can also be generalized to curves in higher dimensional spaces by flattening out the tangent developable to γ into a plane, and computing the total curvature of the resulting curve. That is, the total curvature of a curve in n-dimensional space is
Since these parameterizing functions are doubly periodic, the elliptic curve can be identified with a period parallelogram with the sides glued together i.e. a torus. So the genus of an elliptic curve is 1. The genus–degree formula is a generalization of this fact to higher genus curves. The basic idea would be to use higher degree equations.
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 ...
The field of real numbers is not algebraically closed, the geometry of even a plane curve C in the real projective plane. Assuming no singular points, the real points of C form a number of ovals, in other words submanifolds that are topologically circles. The real projective plane has a fundamental group that is a cyclic group with two elements ...