of the curve also has the same two lines as asymptotes. Although the definition here uses a parameterization of the curve, the notion of asymptote does not depend on the parameterization. In fact, if the equation of the line is then the distance from the point A(t) = (x(t),y(t)) to the line is given by if γ(t) is a change of parameterization then the distance becomes which tends to zero simultaneously as the previous expression. An important case is when the curve is the graph of a real function (a function of one real variable and returning real values). The graph of the function y = ƒ(x) is the set of points of the plane with coordinates (x,ƒ(x)). For this, a parameterization is This parameterization is to be considered over the open intervals (a,b), where a can be −∞ and b can be +∞. An asymptote can be either vertical or non-vertical (oblique or horizontal). In the first case its equation is x = c, for some real number c. The non-vertical case has equation , where m and are real numbers. All three types of asymptotes can be present at the same time in specific examples. Unlike asymptotes for curves that are graphs of functions, a general curve may have more than two non-vertical asymptotes, and may cross its vertical asymptotes more than once. Curvilinear asymptotes Let be a parametric plane curve, in coordinates A(t) = (x(t),y(t)), and B be another (unparameterized) curve. Suppose, as before, that the curve A tends to infinity. The curve B is a curvilinear asymptote of A if the shortest distance from the point A(t) to a point on B tends to zero as t → b. Sometimes B is simply referred to as an asymptote of A, when there is no risk of confusion with linear asymptotes. For example, the function has a curvilinear asymptote , which is known as a parabolic asymptote because it is a parabola rather than a straight line. Asymptotes and curve sketching Asymptotes are used in procedures of curve sketching. An asymptote serves as a guide line to show the behavior of the curve towards infinity. In order to get better approximations of the curve, curvilinear asymptotes have also been used although the term asymptotic curve seems to be preferred. Algebraic curves The asymptotes of an algebraic curve in the affine plane are the lines that are tangent