Algebraic Curve - in Euclidean Geometry

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 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 may be uniquely decomposed into a finite numbers 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 a 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 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.