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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.
An example is the Fermat curve u n + v n = w n, which has an affine form x n + y n = 1. A similar process of homogenization may be defined for curves in higher dimensional spaces. Except for lines, the simplest examples of algebraic curves are the conics, which are nonsingular curves of degree two and genus zero.
1.1 Algebraic curves. 1.1.1 Rational curves. 1.1.1.1 Degree 1. ... This is a list of Wikipedia articles about curves used in different fields: mathematics ...
The twisted cubic is a projective algebraic variety. Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in ...
Examples of the most studied classes of algebraic varieties are lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation.
In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly x and y. For example, the unit circle is defined by the implicit equation + =. In general, every implicit curve is defined by an equation of the form (,) =
Many curves can be defined in either fashion, but the two definitions may not agree. For example, the cusp can be defined on an algebraic curve, =, or on a parametrised curve, () = (,). Both definitions give a singular point at the origin.
In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane.The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves.
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