# Special cubics that are determined by 6 or 7 points

In the case of quadratic curves it is widely known that a conic is uniquely determined by 5 points. But if we use only three points we can uniquely determine a circle and if four points -- a parabola. Both are special curves in some sense (the first one is special in Euclidean geometry and the other one is special in Affine/Projective geometry).

The question is probably a little bit vague, but what special types of cubics can be determined by just 6 or 7 points?

A circle is a conic through the two ideal circle points $[1:\pm i:0]$, so this is just another case of a conic through five points, with two chosen in a special way. You could simply generalize this to cubics by taking these ideal circle points as points on the cubic as well. Since they play a very central part in modelling Euclidean geometry in the projective plane, the result will likely have interesting Euclidean properties as well, but I don't know which.