Lecture 7: Intro to Geometry, Splines, and Bezier Curves (28)
NKJEW
I think it's kind of interesting how this method (and the next) both only require that the function and its derivatives be continuous, but not higher order derivatives. In practice, this is obvious, because adding these extra constraints generally results in much "spikier" curves (overfitting), but the fact that these curves aren't what we're looking for seems more like a side effect of the way we intend to "design" these curves. Namely, what we want is for the curve between any pair of points to be as "simple" as possible, but the state of the curve as a whole is more dependent on where we put the points relative to each other.
I think it's kind of interesting how this method (and the next) both only require that the function and its derivatives be continuous, but not higher order derivatives. In practice, this is obvious, because adding these extra constraints generally results in much "spikier" curves (overfitting), but the fact that these curves aren't what we're looking for seems more like a side effect of the way we intend to "design" these curves. Namely, what we want is for the curve between any pair of points to be as "simple" as possible, but the state of the curve as a whole is more dependent on where we put the points relative to each other.