Pascal’s Theorem states that for six distinct points on a two dimensional conic section forming a hexagon, the intersections of opposite sides are collinear. I am not an expert in geometry, but I thought it would be interesting to try and generalize this theorem to higher dimensions. For example, taking six distinct random points on the surface of an ellipsoid, can an interesting analogous result be found? Clearly in general the opposite sides of the hexagon will not intersect. However, after plotting some examples using Mathematica with six random points on an ellipsoid, I can see that the projection of the hexagon sides onto the the plane of my viewing reference, the projection intersections are always collinear no matter the orientation of the ellipsoid. In the example below I have plotted six random points and varied the viewing angle of the ellipsoid. In each case the intersecting points of the projection to my viewing plane are collinear. The same holds true for sample hyperboloids and sample paraboloids. Pascal's Theorem is a special case of this more general conjecture. Is there a proof of this conjecture, or else some ideas on how to prove this?
The projection of six points on the ellipsoid is six points inside the resulting ellipse. No further constraints apply: any point within the ellipse can be translated back (in a non-unique fashion) to a point on the ellipsoid. Furthermore, any configuration of points in the plane can be enclosed in the projection of an ellipsoid. So your conjecture would imply that any configuration of 6 points were to satisfy Pascal's Theorem, which is clearly not the case.
I'm not sure what happened in your observations. It might well be that speaking in terms of probabilities, configurations where the three intersections in the projection appear almost collinear are somewhat more probable than for 6 random points in the plane, but that would be a different question.