How many points determine a cubic curve?

Question

There are nine coefficients in an equation of cubic curve. It means nine points can determine a unique cubic curve. But two cubic curves A, B can have nine intersection points. If we use the nine intersection points to draw a cubic curve, then why do we get at least two cubic curves,not a unique one?

Answer 1

I came here with the same question. Seeing that there is still no answer, I played around in Mathematica and this is my understanding so far.

There are 10 coefficients in an equation for a cubic. Let us first consider the general case. In a general case, all coefficients are non-zero, and we can normalize the constant to be, say, 1. Then 9 points give us a system of linear equations with rank 9. It will have a unique solution defining a unique cubic curve.

If we choose our 9 points as the intersection points of two distinct cubic curves, we are no longer in the general case. Here we have a system of linear equations that admits two different solutions, which can only happen if the rank is less than 9. Indeed, if I try it in Mathematica, I get a system with rank 8. Of course, then there are infinitely many solutions. Or, in other words, we can take an arbitrary 10th point, and draw a curve that passes through the initial 9 points plus the new point.