# cardinal spline as set of b-splines

Is a cardinal cubic b-spline just a collection of cubic polynomials (each polynomial connecting two points, and each polynomial defined by those two points and their 2 neighbours)?

If so, then can i express a cardinal cubic b-spline as a collection of cubic b-splines, each defined by four control points?

Apparently the first and fourth control points correspond to the two points being connected. But how do i calculate the coordinates of the other two control points (taking into account the tension value). Also, what happens at the endpoints of the curve?

Is a cardinal cubic b-spline just a collection of cubic polynomials

Yes. Actually any cubic spline is just a string of polynomial cubic segments joined end-to-end. This latter is called the "piecewise polynomial" representation of the spline. Any spline can be represented in piecewise polynomial form.

can i express a cardinal cubic b-spline as a collection of cubic b-splines, each defined by four control points?

Yes, you can take each of the polynomial cubic segments mentioned above, and express it in any basis you like. A polynomial cubic segment represented by four b-spline control points (with the knot sequence 0,0,0,0,1,1,1,1) is called a Bezier curve.

Apparently the first and fourth control points correspond to the two points being connected.

Correct.

But how do i calculate the coordinates of the other two control points

Since you know the polynomials, you can calculate first derivative $U$ at the start-point $P_0$ and the first derivative $V$ at the end-point $P_3$. The two "interior" control points are then $P_1 = P_0 + \tfrac13U$ and $P_2 = P_3 - \tfrac13V$. A spline segment represented by end-points and end derivatives in this way is said to be in "Hermite" form.