Mathematical Structures for Computer Graphics

7.3 B-splines

Designing a curve is not simple. The Bézier approach helps significantly, and the key idea is that the curve is not forced to go through all the control points. Still, moving a control point affects the shape of the entire curve and often we need to adjust only a small portion. The optimal balance between global and local control is hard to achieve in all situations. One good technique is to build curves in pieces hoping that the pieces can be put together in a reasonably smooth way.

Before the widespread use of computer techniques in design, those doing drafting for architectural drawings often used thin strips of wood or metal to shape a curve and then trace it onto paper. These splines have a natural smoothness and a minimum amount of fluctuations. It is not entirely easy to adjust only small regions of the spline, but if they are thin enough, the perturbations can be isolated to smaller lengths of the whole strip.

Attempts to mathematically model the flexibility of the spline by incorporating the relevant physics become complicated, and, instead, several approximations have been proposed emphasizing some aspects of the physical spline over others. Useful directions usually describe the curve as an affine combination of control points with polynomial or rational blending functions plus some extra constraints (like specified tangents). Curve segments are then put together piecewise to form a larger easy-to-alter curve. Several mathematicians in the past have ...