[lucas-beyer]

No.

As soon as you have 4 points, these 4 points uniquely define a cubic. Even if those 4 points are "right next" to eachother.

If this is difficult to imagine, it is the same with a line. Wherever you place 2 points, they always uniquely define a single line.

Unless, of course, you place 2 points exactly on top of eachother.

[xk_id]

really? that's amazing! I'd love to see a formal proof of this.

edit: I suppose this counts as proof http://www.had2know.com/academics/cubic-through-4-points.htm..., but atm I don't remember why matrixes are connected with equations.

[joshzayin]

There's a more general proof at https://docs.google.com/viewer?a=v&q=cache:euFgQLUCTfwJ:... that looks reasonable at a skim.

[xk_id]

Awesome!