Editing Ellipse (section)

=== Steiner generation ===
[[File:Ellipse-steiner-e.svg|250px|thumb|Ellipse: Steiner generation]]
[[File:Ellipse construction - parallelogram method.gif|200px|thumb|Ellipse: Steiner generation]]
The following method to construct single points of an ellipse relies on the [[Steiner conic|Steiner generation of a conic section]]:
: 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 ellipse <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 = (0,\, b)</math> be an upper co-vertex of the ellipse and <math>A = (-a,\, 2b),\, B = (a,\,2b)</math>.

<math>P</math> is the center of the rectangle <math>V_1,\, V_2,\, B,\, A</math>. The side  <math>\overline{AB}</math> of the rectangle is divided into n equal spaced line segments and this division is projected parallel with the diagonal <math>AV_2</math> as direction onto the line segment  <math>\overline{V_1B}</math> and assign the division as shown in the diagram. The parallel projection together with the reverse of the orientation 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>V_1 B_i</math> and <math>V_2 A_i</math> are points of the uniquely defined ellipse. With help of the points <math>C_1,\, \dotsc</math> the points of the second quarter of the ellipse can be determined. Analogously one obtains the points of the lower half of the ellipse.

Steiner generation can also be defined for hyperbolas and parabolas. It 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.