Editing Hyperbola (section)

====Point construction====
[[File:Hyperbel-pasc4-s.svg|thumb|Point construction: asymptotes and ''P''<sub>1</sub> are given → ''P''<sub>2</sub>]]

For a hyperbola with parametric representation <math>\vec x = \vec p(t) = \vec f_1 t + \vec f_2 \tfrac{1}{t}</math> (for simplicity the center is the origin) the following is true:

{{block indent |em=1.5 |text=For any two points <math>P_1:\ \vec f_1 t_1+ \vec f_2 \tfrac{1}{t_1},\ P_2:\ \vec f_1 t_2+ \vec f_2 \tfrac{1}{t_2}</math> the points
<math display="block">A:\ \vec a =\vec f_1 t_1+ \vec f_2 \tfrac{1}{t_2}, \ B:\ \vec b=\vec f_1 t_2+ \vec f_2 \tfrac{1}{t_1}</math>
are collinear with the center of the hyperbola (see diagram).}}
The simple proof is a consequence of the equation <math>\tfrac{1}{t_1}\vec a = \tfrac{1}{t_2}\vec b</math>.

This property provides a possibility to construct points of a hyperbola if the asymptotes and one point are given.

This property of a hyperbola is an affine version of the 4-point-degeneration of [[Pascal's theorem]].<ref>[https://www2.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf Lecture Note ''Planar Circle Geometries'', an Introduction to Moebius-, Laguerre- and Minkowski Planes], S.&nbsp;32, (PDF; 757&nbsp;kB)</ref>