Editing Hyperbola (section)

====Tangent construction====
[[File:Hyperbel-tang-s.svg|thumb|Tangent construction: asymptotes and ''P'' given → tangent]]
The tangent vector can be rewritten by factorization:
<math display="block">\vec p'(t)=\tfrac{1}{t}\left(\vec f_1t - \vec f_2 \tfrac{1}{t}\right) \ .</math>
This means that

{{block indent |em=1.5 |text=the diagonal <math>AB</math> of the parallelogram <math>M: \ \vec f_0, \ A=\vec f_0+\vec f_1t,\ B:\ \vec f_0+ \vec f_2 \tfrac{1}{t},\ P:\ \vec f_0+\vec f_1t+\vec f_2 \tfrac{1}{t}</math> is parallel to the tangent at the hyperbola point <math>P</math> (see diagram).}}

This property provides a way to construct the tangent at a point on the hyperbola.

This property of a hyperbola is an affine version of the 3-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;33, (PDF; 757&nbsp;kB)</ref>

;Area of the grey parallelogram:
The area of the grey parallelogram <math>MAPB</math> in the above diagram is
<math display="block">\text{Area} = \left|\det\left( t\vec f_1, \tfrac{1}{t}\vec f_2\right)\right| = \left|\det\left(\vec f_1,\vec f_2\right)\right| = \cdots = \frac{a^2+b^2}{4} </math>
and hence independent of point <math>P</math>. The last equation follows from a calculation for the case, where <math>P</math> is a vertex and the hyperbola in its canonical form <math>\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}=1 \, .</math>