How do Catmull-Rom and Hermite splines relate?

Question

I'm a little confused as to how they relate. From my class, my professor asked how to fit C1 continuous piecewise Hermite curves to x amount of points with automatically computed tangents. Can someone explain how this works?

Answer 1

Hermite spline is a method of representation of cubic curve with two endpoints and two tangent vectors at these endpoints.
Note that there are many approaches to represent the same curve- power basis (at^3+bt^2+ct+d), Bernstein polynomials (for Bezier curves) and so on.

Catmull-Rom spline is a method of construction of a cubic curve (C1 continuous, if some pieces used). Every Catmull-Rom segment is a Hermite spline. If we want to represent a Catmull-Rom spline for the P1-P2 segment of the (P0,P1,P2,P3) point sequence as a Hermite one, we just use P1 and P2 as endpoints, and V1 and V2 as tangent vectors, where

V1 = tau * (P2-P0) 
V2 = tau * (P3-P1) 

(good article here) and tau is parameter of tension (rigidness)