ConicCharacterizationEllipse

Conic characterization through a rational function, ellipse - parabola

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 . If the denominator has no or one real root, then the corresponding conic is respectively an ellipse or a parabola.
In the first case, absorbing some multipliers into vectors (e, E), the function can be brought into the form f(t) = (t-a)/((t-b)²+c²).
In the second case set simply c=0, thus obtaining f(t) = (t-a)/(t-b)².
There is a degeneration possible when c=0, b=a, in which case, f(t) = 1/(t-a).
Then set S coincides with the set of points of the line x-ay=1 whose direction is that of the tangent at O and passes through e.

Conic characterization through a rational function, ellipse - parabola

See Also