I'm developing an optimization problem that requires me to quantify the "curviness" of a spline. The spline is defined in a software library, and the only input to generate a spline with this library is a set of control points. The spline is then parametrized in the range [0, 1], and the software library allows me to then calculate the position, tangent, and normal at any point t along the range.
Given this, how might I be able to quantify how curvy the spline is between two adjacent control points (e.g., if the spline through the two control points is simply a straight line, then the metric should give a value of 0, but if it is curved, anything > 0).
This might be a good place to start. It discusses curves that minimize "energy" and variation of curvature, which are both plausible measures of curviness. If you like Levien's approach, you can find a lot more detail in his Ph.D. thesis.