Editing Hyperbola (section)

===Steiner generation of a hyperbola===
[[File:Hyperbel-steiner-e.svg|250px|thumb|Hyperbola: Steiner generation]]
[[File:Hyperbola construction - parallelogram method.gif|200px|thumb|Hyperbola ''y'' = 1/''x'': Steiner generation]]
The following method to construct single points of a hyperbola relies on the [[Steiner conic|Steiner generation of a non degenerate conic section]]:

{{block indent |em=1.5 |text=Given two [[pencil (mathematics)|pencils]] <math>B(U),B(V)</math> of lines at two points <math>U,V</math> (all lines containing <math>U</math> and <math>V</math>, respectively) and a projective but not perspective mapping <math>\pi</math> of <math>B(U)</math> onto <math>B(V)</math>, then the intersection points of corresponding lines form a non-degenerate projective conic section.}}

For the generation of points of the hyperbola <math>\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2} = 1</math> one uses the pencils at the vertices <math>V_1,V_2</math>. Let <math>P = (x_0,y_0)</math> be a point of the hyperbola and <math>A = (a,y_0), B = (x_0,0)</math>. The line segment <math>\overline{BP}</math> is divided into n equally-spaced segments and this division is projected parallel with the diagonal <math>AB</math> as direction onto the line segment <math>\overline{AP}</math> (see diagram). The parallel projection is part of the projective mapping between the pencils at <math>V_1</math> and <math>V_2</math> needed. The intersection points of any two related lines <math>S_1 A_i</math> and <math>S_2 B_i</math> are points of the uniquely defined hyperbola.

''Remarks:''
* The subdivision could be extended beyond the points <math>A</math> and <math>B</math> in order to get more points, but the determination of the intersection points would become more inaccurate. A better idea is extending the points already constructed by symmetry (see animation).
* The Steiner generation exists for ellipses and parabolas, too.
* The Steiner generation is sometimes called a ''parallelogram method'' because one can use other points rather than the vertices, which starts with a parallelogram instead of a rectangle.