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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. With a curve given by such an implicit equation, the first problems are 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 in solving these problems. Every algebraic curve may be uniquely decomposed into a finite number of smooth monotone arcs also called branches sometimes connected by some points sometimes called "remarkable points", and possibly a finite number of isolated points called acnodes. 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 usually called an infinite arc or has an endpoint 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 , there are two infinite arcs having the origin 0,0 as of endpoint.
This point is the only singular point of the curve. There are also two arcs having this singular point as one endpoint and having a second endpoint with a horizontal tangent. Finally, there are two other arcs each having one of these points with horizontal tangent as the first endpoint and having the unique point with vertical tangent as the second endpoint.
In contrast, 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 asymptotes if any and the way in which the arcs connect them. It is also useful to consider the inflection points as remarkable points. When all this information is drawn on a sheet of paper, the shape of the curve usually appears 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 the section Projective curves.
Plane projective curves[ edit ] It is often desirable to consider curves in the projective space. An algebraic curve in the projective plane or plane projective curve is the set of the points in a projective plane whose projective coordinates are zeros of a homogeneous polynomial in three variables P x, y, z.
Plane Algebraic Curves
Plane Algebraic Curves: Translated by John Stillwell