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Real Plane Curve
In mathematics, a real plane curve is usually a real algebraic curve defined in the real projective plane. Ovals 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. Such an oval may represent either group element; in other words we may or may not be able to contract it down in the plane. Taking out the line at infinity ''L'', any oval that stays in the finite part of the affine plane will be contractible, and so represent the identity element of the fundamental group; the other type of oval must therefore intersect ''L''. There is still the question of how the various ovals are nested. This was the topic of Hilbert's sixteenth problem. See Harnack's curve theorem for a classi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
