Editing Parabola (section)

== Steiner generation ==
=== Parabola ===
[[File:Parabel-steiner-s.svg|thumb|Steiner generation of a parabola]]

[[Jakob Steiner|Steiner]] established the following procedure for the construction of a non-degenerate conic (see [[Steiner conic]]):
{{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>, the intersection points of corresponding lines form a non-degenerate projective conic section.}}

This procedure can be used for a simple construction of points on the parabola <math>y = ax^2</math>:
* Consider the pencil at the vertex <math>S(0, 0)</math> and the set of lines <math>\Pi_y</math> that are parallel to the ''y'' axis.
*# Let <math>P = (x_0, y_0)</math> be a point on the parabola, and <math>A = (0, y_0)</math>, <math>B = (x_0, 0)</math>.
*# The line segment <math>\overline{BP}</math> is divided into ''n'' equally spaced segments, and this division is projected (in the direction <math>BA</math>) onto the line segment <math>\overline{AP}</math> (see figure). This projection gives rise to a projective mapping <math>\pi</math> from pencil <math>S</math> onto the pencil <math>\Pi_y</math>.
*# The intersection of the line <math>SB_i</math> and the ''i''-th parallel to the ''y'' axis is a point on the parabola.

''Proof:'' straightforward calculation.

''Remark:'' Steiner's generation is also available for [[ellipse]]s and [[hyperbola]]s.

=== Dual parabola ===
[[File:Parabel-bezier.svg|400px|thumb|Dual parabola and Bézier curve of degree 2 (right: curve point and division points <math>Q_0, Q_1</math> for parameter <math>t = 0.4</math>)]]

A ''dual parabola'' consists of the set of tangents of an ordinary parabola.

The Steiner generation of a conic can be applied to the generation of a dual conic by changing the meanings of points and lines:
{{block indent | em = 1.5 | text = Let be given two point sets on two lines <math>u, v</math>, and a projective but not perspective mapping <math>\pi</math> between these point sets, then the connecting lines of corresponding points form a non degenerate dual conic.}}

In order to generate elements of a dual parabola, one starts with
# three points <math>P_0, P_1, P_2</math> not on a line,
# divides the line sections <math>\overline{P_0 P_1}</math> and <math>\overline{P_1 P_2}</math> each into <math>n</math> equally spaced line segments and adds numbers as shown in the picture.
# Then the lines <math>P_0 P_1, P_1 P_2, (1,1), (2,2), \dotsc</math> are tangents of a parabola, hence elements of a dual parabola.
# The parabola is a [[Bézier curve]] of degree 2 with the control points <math>P_0, P_1, P_2</math>.

The ''proof'' is a consequence of the ''[[de Casteljau algorithm]]'' for a Bézier curve of degree 2.