I understand how the Hermite basis functions are derived, but I'm confused about why we derived them at all? Is it just an alt. basis that reintroduces ttt into the interpolation?
I think so, and it has to do with evaluating P'(0) and P'(1) as opposed to interpolating a cubic from four points
@ellinzhao: Having the Hermite basis functions precomputed for values of t allows you to save on time when interpolating points, since you can just multiply those values into the control points and their derivatives and add to get the interpolation (versus having to solve the whole system of equations each time)