Advantages of cubic spline interpolation over cubic Hermite interpolation? So far I have studied cubic Hermite interpolation has a less computational cost and does not have issues like undershoot and overshoot as compared to cubic spline interpolation. My question is what are the advantages of cubic spline interpolation over cubic Hermite interpolation? And in what situation/data application of cubic spline interpolation will be more suitable?
 A: Cubic spline interpolation of a function $f : [a,b] \rightarrow \mathbb{R}$ produces an approximation $p : [a,b] \rightarrow \mathbb{R}$ which is two times differentiable and $p''$ is continuous. This is particularly useful when you need to fool the human eye. In the cinema the audience expects the spaceship's position, velocity and acceleration to wary continuously as it moves across the screen.
Cubic Hermite interpolation produces an approximation $q : [a,b] \rightarrow \mathbb{R}$ that is differentiable with a continuous derivative. Cubic Hermite interpolation is useful when generating continuous output when solving ordinary differential equations $$x'(t) = g(t,x(t)).$$ It uses the local position ($x$) and velocity data ($x'$) that is naturally available in this context and it matches the accuracy delivered by the classical 4th order Runge Kutta method. Cubic Hermite interpolation allows us to rapidly solve event equations of the type $$h(x(t),x'(t)) = 0.$$ This allows us to determine when, say, an artillery shell reaches the apex of its trajectory or impacts the ground.
