The parametric second order curve
r = (a0 + a1 lambda + a2 lambda lambda)/(alpha + lambda lambda),
with a0, a1 and a2 three arbitrary vectors and alpha, lambda scalars, represents for
alpha = 0 a parabola.
With the animated variation of lambda (orange line) the yellow radius vector sweeps out the parabola. In the spherical view it becomes clear that the parabola is a limit to the ellipse, an ellipse, which closes in one point at infinity. Infinity in this view is on the blue equator of the sphere.
It may take some minutes until the graph below appears!
Compare the Euclidean view.
Soli Deo Gloria. Created with Cinderella by Eckhard Hitzer (Fukui).