ConicCharacterizationHyperbola

Conic characterization through a rational function, hyperbola

Given are two independent vectors {E, e} and a rational function of the form.
f(t) = (at+b)/(ct²+dt+e).
The set of points S={f(t)(E+te)} describes parametrically a conic passing through the origin of coordinates.
Inversely, every conic passing through the origin of coordinates and every set of two independent vectors {E, e} defines such a function f(t) so that the corresponding set S coincides with the original conic.

This has been proved in ConicCharacterization.html . In ConicCharacterizationEllipse.html has been handled the case of ellipse and parabola.
Here I study the case of the hyperbola, in which the denominator has two different real roots, and the corresponding conic is a hyperbola.
In this case, absorbing some multipliers into vectors (e, E), the function can be brought into the form f(t) = (t-a)/((t-b)*(t-c)).
There is a degeneration possible when a=b or a=c. Then set S coincides with the set of points of a line (see ConicCharacterizationEllipse.html ).

The important thing in this case is the existence of the asymptotes. Their direction is obtained through vectors.
B = E + be and C = E + ce.
The figure suggests a way to find the center Q of the conic using the middle t₃/2 and the intersection of the tangents at O and t₃ (by t₃ I mean point t₃E).

Conic characterization through a rational function, hyperbola

See Also