Editing Asymptote (section)

==Algebraic curves==
[[File:Folium Of Descartes.svg|thumb|right|A [[cubic curve]], [[Folium of descartes|the folium of Descartes]] (solid) with a single real asymptote (dashed)]]
The asymptotes of an [[algebraic curve]] in the [[affine space|affine plane]] are the lines that are tangent to the [[algebraic curve#Projective curves|projectivized curve]] through a [[point at infinity]].<ref>C.G. Gibson (1998) ''Elementary Geometry of Algebraic Curves'', § 12.6 Asymptotes, [[Cambridge University Press]] {{ISBN|0-521-64140-3}},</ref> For example, one may identify the [[unit hyperbola#Asymptotes|asymptotes to the unit hyperbola]] in this manner. Asymptotes are often considered only for real curves,<ref>{{Citation | last1=Coolidge | first1=Julian Lowell | title=A treatise on algebraic plane curves | publisher=[[Dover Publications]] | location=New York | mr=0120551 | year=1959 | isbn=0-486-49576-0}}, pp. 40&ndash;44.</ref> although they also make sense when defined in this way for curves over an arbitrary [[field (mathematics)|field]].<ref>{{Citation | last1=Kunz | first1=Ernst | title=Introduction to plane algebraic curves | publisher=Birkhäuser Boston | location=Boston, MA | isbn=978-0-8176-4381-2 | mr=2156630 | year=2005}}, p. 121.</ref>

A plane curve of degree ''n'' intersects its asymptote at most at ''n''&minus;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'')&nbsp;=&nbsp;0 where ''P'' is a polynomial of degree ''n''
:<math>P(x,y) = P_n(x,y) + P_{n-1}(x,y) + \cdots + P_1(x,y) + P_0</math>
where ''P''<sub>''k''</sub> is [[homogeneous polynomial|homogeneous]] of degree ''k''.  Vanishing of the linear factors of the highest degree term ''P''<sub>''n''</sub> defines the asymptotes of the curve: setting {{math|1=''Q'' = ''P''<sub>''n''</sub>}}, if {{math|1=''P''<sub>''n''</sub>(''x'', ''y'') = (''ax'' &minus; ''by'') ''Q''<sub>''n''&minus;1</sub>(''x'', ''y'')}}, then the line
:<math>Q'_x(b,a)x+Q'_y(b,a)y + P_{n-1}(b,a)=0</math>
is an asymptote if <math>Q'_x(b,a)</math> and <math>Q'_y(b,a)</math> are not both zero. If  <math>Q'_x(b,a)=Q'_y(b,a)=0</math> and <math>P_{n-1}(b,a)\neq 0</math>, there is no asymptote, but the curve has a branch that looks like a branch of parabola. Such a branch is called a '''{{vanchor|parabolic branch}}''', even when it does not have any parabola that is a curvilinear asymptote. If <math>Q'_x(b,a)=Q'_y(b,a)=P_{n-1}(b,a)=0,</math> the curve has a singular point at infinity which may have several asymptotes or parabolic branches.

Over the complex numbers, ''P''<sub>''n''</sub> splits into linear factors, each of which defines an asymptote (or several for multiple factors). Over the reals, ''P''<sub>''n''</sub> 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 {{nowrap|1=''x''<sup>4</sup> + ''y''<sup>2</sup> - 1 = 0}} has no real points outside the square <math> |x|\leq 1, |y|\leq 1</math>, but its highest order term gives the linear factor ''x'' with multiplicity 4, leading to the unique asymptote ''x''=0.