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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.
A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form (,) = for some specific function f.If this equation can be solved explicitly for y or x – that is, rewritten as = or = for specific function g or h – then this provides an alternative, explicit, form of the representation.
In classical algebraic geometry, the genus–degree formula relates the degree of an irreducible plane curve with its arithmetic genus via the formula: = (). Here "plane curve" means that is a closed curve in the projective plane.
Download as PDF; Printable version; ... Algebraic curves (7 C, 95 P) C. ... (1 C, 16 P) Pages in category "Plane curves" The following 45 pages are in this category ...
The cruciform curve, or cross curve is a quartic plane curve given by the equation = where a and b are two parameters determining the shape of the curve. The cruciform curve is related by a standard quadratic transformation, x ↦ 1/x, y ↦ 1/y to the ellipse a 2 x 2 + b 2 y 2 = 1, and is therefore a rational plane algebraic curve of genus zero.
The Trott curve and seven of its bitangents. The others are symmetric with respect to 90° rotations through the origin. The Trott curve with all 28 bitangents. In the theory of algebraic plane curves, a general quartic plane curve has 28 bitangent lines, lines that are tangent to the curve in two
Another source of complexity is the generality of Kempe's application to all algebraic curves. By focusing on parameterized algebraic curves, dual quaternion algebra can be used to factor the motion polynomial and obtain a drawing linkage. [8] This has been extended to provide movement of the end-effector, but again for parameterized curves. [9]
In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane P 2. The original form states: Assume that two cubics C 1 and C 2 in the projective plane meet in nine (different) points, as they do in general over an algebraically closed field. Then every cubic that passes ...