Second order curves are conics

The parametric second order curves
r = (a0 + a1 lambda + a2 lambda lambda)/(alpha + lambda lambda),
with a0, a1 and a2 three arbitrary vectors and alpha, lambda scalars, represent all conics:

* alpha > 0 ... ellipse
* alpha = 0 ... parabola
* alpha < 0 ... hyperbola

I recommend to first change alpha to see the different conics, then change lambda, to see how the yellow radius vector r draws the conics and finally to change the three vectors a0, a1, a2, to see how they affect shape and orientation. You can change the position of all bright red points interactively. Especially after changing a0, a1, a2 or alpha, it may take a few minutes until Cinderella generates the new conic.

Please enable Java for an interactive construction (with Cinderella).

[ conics | GA with Cinderella ]

Soli Deo Gloria. Created with Cinderella by Eckhard Hitzer (Fukui).