Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 77 / 4 / iv

A practical family consists of constant-distance parallel surfaces and smooth variable normal offsetsConstants cover wall thickness, cutter radius and clearance. A scalar B-spline or piecewise polynomial displacement field permits thickness variation, local bumps and smooth blending; compact support makes local edits economical. Use the base parameterization or compatible data on its subdivision hierarchy so that evaluating the displacement and its derivatives requires only a small local stencil. Choose displacement size and gradients to avoid unintended folds, and trim global self-intersections where the physical application requires an envelope. A constant displacement smaller than the local curvature radii prevents the corresponding local focal singularities, although global clearance still needs a separate check.

When remaining within a polynomial or rational representation is more useful than prescribing an exact distance, a related practical form is $P+\lambda (u,v)(P_u\times P_v)$ with a low-degree scalar field $\lambda$. Its displacement is normal but has magnitude $|\lambda |\Vert P_u\times P_v\Vert$, so it is not a constant-distance parallel surface. For polynomial $P$ and $\lambda$, it stays polynomial; for rational input it stays rational. More general low-degree vector displacement fields similarly preserve efficient evaluation, while relinquishing the normal-offset condition. These forms make explicit the tradeoff between an exact metric offset, which involves normalization, and an easily represented deformation. Simple scalar normal-displacement fields are useful and directly evaluable; algebraic displacement fields are useful when representation closure is required.