Editing Asymptote (section)

==Asymptotic cone==
[[File:Conic section hyperbola.gif|thumb|Hyperbolas, obtained cutting the same right circular cone with a plane and their asymptotes]]
The [[hyperbola]]
:<math>\frac{x^2}{a^2}-\frac{y^2}{b^2}= 1</math>
has the two asymptotes
:<math>y=\pm\frac{b}{a}x.</math>
The equation for the union of these two lines is
:<math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=0.</math>
Similarly, the [[hyperboloid]]
:<math>\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1</math>
is said to have the '''asymptotic cone'''<ref>[https://books.google.com/books?id=YMU0AAAAMAAJ L.P. Siceloff, G. Wentworth, D.E. Smith ''Analytic geometry'' (1922) p. 271]</ref><ref>[https://books.google.com/books?id=fGg4AAAAMAAJ P. Frost ''Solid geometry'' (1875)] This has a more general treatment of asymptotic surfaces.</ref>

:<math>\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=0.</math>

The distance between the hyperboloid and cone approaches 0 as the distance from the origin approaches infinity.

More generally, consider a surface that has an implicit equation
<math>P_d(x,y,z)+P_{d-2}(x,y,z) + \cdots P_0=0,</math>
where the <math>P_i</math> are [[homogeneous polynomial]]s of degree <math> i </math> and <math>P_{d-1}=0</math>. Then the equation <math>P_d(x,y,z)=0</math> defines a [[cone]] which is centered at the origin. It is called an '''asymptotic cone''', because the distance to the cone of a point of the surface tends to zero when the point on the surface tends to infinity.