page_title: Asymptote
section_title: Algebraic curves
hierarchical_section_title: Asymptote / Algebraic curves
context_page_description: In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity.
  The word asymptote is derived from the Greek ἀσύμπτωτος which means "not falling together", from ἀ priv. + σύν "together" + πτωτ-ός "fallen". The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve.
  There are three kinds of asymptotes: horizontal, vertical and oblique. For curves given by the graph of a function y = ƒ(x), horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞. Vertical asymptotes are vertical lines near which the function grows without bound. An oblique asymptote has a slope that is non-zero but finite, such that the graph of the function approaches it as x tends to +∞ or −∞.
context_section_description: The asymptotes of an algebraic curve in the affine plane are the lines that are tangent to the projectivized curve through a point at infinity. For example, one may identify the asymptotes to the unit hyperbola in this manner. Asymptotes are often considered only for real curves, although they also make sense when defined in this way for curves over an arbitrary field.
  A plane curve of degree n intersects its asymptote at most at n−2 other points, by Bézout's theorem, as the intersection at infinity is of multiplicity at least two.  For a conic, there are a pair of lines that do not intersect the conic at any complex point: these are the two asymptotes of the conic.
  A plane algebraic curve is defined by an equation of the form P(x,y) = 0 where P is a polynomial of degree n
  where Pₖ is homogeneous of degree k.  Vanishing of the linear factors of the highest degree term Pₙ defines the asymptotes of the curve: setting Q = Pₙ, if Pₙ(x, y) = (ax − by) Qₙ−1(x, y), then the line
  is an asymptote if  and  are not both zero. If   and , there is no asymptote, but the curve has a branch that looks like a branch of parabola. Such a branch is called a parabolic branch, even when it does not have any parabola that is a curvilinear asymptote. If  the curve has a singular point at infinity which may have several asymptotes or parabolic branches.
  Over the complex numbers, Pₙ splits into linear factors, each of which defines an asymptote (or several for multiple factors).  0ver the reals, Pₙ splits in factors that are linear or quadratic factors. Only the linear factors correspond to infinite (real) branches of the curve, but if a linear factor has multiplicity greater than one, the curve may have several asymptotes or parabolic branches. It may also occur that such a multiple linear factor corresponds to two complex conjugate branches, and does not corresponds to any infinite branch of the real curve.  For example, the curve x⁴ + y² - 1 = 0 has no real points outside the square , but its highest order term gives the linear factor x with multiplicity 4, leading to the unique asymptote x=0.
caption: A cubic curve, the folium of Descartes (solid) with a single real asymptote (dashed).