Parabola: the set of points in the plane that are equidistant from a line, called the directrix, and a point, called the focal point, not on the line.
Suppose we try to replicate this on a sphere: let the directrix be given as a great circle $D$. Let $C$ be a great circle perpendicular to $D$. Let $P$ be our focal point, lying on $C$ (but not lying on the directrix $D$). Let e be an arbitrary point on $D$, and let $E$ be the great circle passing through e, perpendicular to $D$. Let a great circle through $P$ meet $E$ at point f such that f is equidistance from e and $P$. Finally, let our "parabola" consists of all such points equidistant from focal point $P$ and directrix $D$ (measured along the "straight lines" of great circles).
From what I can tell, there are only two possibilities for the shape of this "parabola": either (1) $P$ will lie on one of the apexes of the circle (that is, one of the two points where great circles perpendicular to $D$ intersect), in which case the "parabola" will be a small circle; otherwise, (2) $P$ will lie at some other point along $C$, in which case I'm uncertain about the shape, other than that it will be symmetrical with respect to $C$ and occupy one hemisphere (since it will lie entirely above $D$).
What will be the shape of the curve under condition (2)?