Cubic B spline drawing shape

Question

When using a cubic B spline with n+1 control points, which shape is difficult to draw / modify ?

Please give an example.

Answer 1

There are theoretical and practical limitations.

First the practical one: if the spline has a large number of control points (more than a dozen or so) then very smooth shapes are difficult to obtain by moving control points around. For example, take a cubic b-spline with around 10 randomly-located control points, and try to move those points to make the curve into a semi-circle (approximately). It can be done, but it requires considerable skill.

So, to make very smooth curves, designers often prefer to use Bezier curves (i.e. b-spline curves with degrees of around 5 or 6 that have only a single segment).

Next, a theoretical problem. No matter how skillful you are, there are certain shapes that can not be represented exactly by a cubic b-spline. For example, circles and other conic sections can not be represented exactly (unless you use rational splines rather than polynomial ones). Also, offsets of spline curves can not be represented as spline curves, except in very special cases. Finally, cubic b-splines are only $C_2$, at best, so they can not represent curves that have higher orders of continuity. In automobile body design, especially, people want curves and surfaces that are $C_3$.

Good approximations of conics and offsets and $C_3$ curves can be constructed, and these are adequate for engineering and manufacturing applications, so the theoretical problems are not very significant in practice.