I need to find the equation of that curve to pass through all and every point (exact fit). I think that to do this I need a polynomial whose grade is equal to the number of points less 1.
Yes, to get a polynomial to pass through 3 points exactly you'll need a quadratic, 4 points you'll need a cubic, etc. You need one variable for each constraint (point to pass through), then it's just a linear system you can solve. However this will of course give you only polynomial curves. If your curve is an ellipse as you said, no polynomial will match it, and if you try this procedure you'll probably get something weird. High-degree polynomials aren't very good approximators, either - see Runge's phenomenon.
If your ultimate goal is to produce an "average" approximating curve, it may well be best to forego the exact-fit polynomials altogether and jump directly to least-squares optimization via Levenberg-Marquardt or similar. This requires you to choose the degree of curve to fit, though (or choose the functional form of the curve more generally). IIRC, polynomial fitting can be done directly by linear least squares, with no need for L-M or more complicated stuff, at least as long as the number of points is not too big. But L-M will work for other kinds of curves too, or for larger numbers of points.