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In mathematics, an algebraic curve or plane algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables. For example, the unit circle is an algebraic curve, being the set of zeros of the polynomial . Various technical considerations result in the complex zeros of a polynomial being considered as belonging to the curve. Also, the notion of algebraic curve has been generalized to allow the coefficients of the defining polynomial and the coordinates of the points of the curve to belong to any field, leading to the following definition. In algebraic geometry, a plane affine algebraic curve defined over a field is the set of points of whose coordinates are zeros of some bivariate polynomial with coefficients in , where is some algebraically closed extension of . The points of the curve with coordinates in are the -points of the curve and, all together, are the part of the curve. For example, is a point of the curve defined by and the usual unit circle is the real part of this curve. The term "unit circle" may refer to all the complex points as well to only the real points, the exact meaning usually clear from the context. The equation defines an algebraic curve, whose real part is empty. More generally, one may consider algebraic curves that are not contained in the plane, but in a space of higher dimension. A curve that is not contained in some plane is called a skew curve. The simplest example of a skew algebraic curve is the twisted cubic. One may also consider algebraic curves contained in the projective space and even algebraic curves that are defined independently to any embedding in an affine or projective space. This leads to the most general definition of an algebraic curve: In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. == In Euclidean geometry == 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, by opposition to the curves that are the graph of a function defining ''explicitly'' ''y'' as a function of ''x''. Given a curve given by such an implicit equation, the first problems that occur is to determine the shape of the curve and to draw it. These problems are not as easy to solve as in the case of the graph of a function, for which ''y'' may easily be computed for various values of ''x''. The fact that the defining equation is a polynomial implies that the curve has some structural properties that may help to solve these problems. Every algebraic curve may be uniquely decomposed into a finite number of smooth monotone arcs (also called ''branches'') connected by some points sometimes called "remarkable points". A ''smooth monotone arc'' is the graph of a smooth function which is defined and monotone on an open interval of the ''x''-axis. In each direction, an arc is either unbounded (one talk of an ''infinite arc'') or has an end point which is either a singular point (this will be defined below) or a point with a tangent parallel to one of the coordinate axes. For example, for the Tschirnhausen cubic of the figure, there are two infinite arcs having the origin (0,0) as end point. This point is the only singular point of the curve. There are two arcs having this singular point as one end point and having a second end point with a horizontal tangent. Finally, there are two other arcs having these points with horizontal tangent as first end point and sharing the unique point with vertical tangent as second end point. On the other hand, the sinusoid is certainly not an algebraic curve, having an infinite number of monotone arcs. To draw an algebraic curve, it is important to know the remarkable points and their tangents, the infinite branches and their asymptote (if any) and the way in which the arcs connect them. It is also useful to consider also the inflection points as remarkable points. When all this information is drawn on a paper sheet, the shape of the curve appears usually rather clearly. If not it suffices to add a few other points and their tangents to get a good description of the curve. The methods for computing the remarkable points and their tangents are described below, after section Projective curves. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Algebraic curve」の詳細全文を読む スポンサード リンク
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