= Solution
A practical family consists of constant-distance <parallel surfaces> and smooth <variable normal offsets>
$$
\boxed{P_d(u,v)=P(u,v)+d(u,v)n(u,v),\qquad d(u,v)\text{ represented by a low-degree scalar spline}.}
$$
Constants 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 $|\lambda|\lVert P_u\times P_v\rVert$, 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. \b[Simple scalar normal-displacement fields are useful and directly evaluable; algebraic displacement fields are useful when representation closure is required.]
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