An important issue here is keeping the number of points interpolated as low as possible. As the degree of the interpolating polynomial
increases, it loses smoothness, and often results in an awful approximation in all but the interpolated points.
A common solution is to approximate different intervals of the interpolated function with different polynomials of low degree, and to impose continuity and differentiability conditions on the whole. Different formulas give us different kinds of curves, like splines, bezier curves, etc.