 Solution

For a regular parametric surface $P(u,v)$, form $N=P_u\times P_v$, $n=N/\Vert N\Vert$ and define the offset evaluatorIt is already a parametric surface with the same parameter domain, even if its coordinates no longer belong to the original polynomial or rational patch family. Point interrogation requires evaluations of $P,P_u,P_v$ and one normalization. For tangent or normal vector interrogation, differentiate the evaluator. For example,and similarly for $v$. The offset normal vector is the normalized cross product $(P_d{)}_u\times (P_d{)}_v$ wherever it is nonzero. These derivatives also support intersection, projection and curvature calculations by the usual methods for parametric surfaces. For a variable normal offset, add $d_{u}n$ and $d_{v}n$ to the corresponding tangent derivatives.

With shape operator $W=-Dn$, the constant-distance offset differential is $D{P}_d=(I-dW)DP$. Along a principal direction its factor is $1-d{\kappa }_i$, where ${\kappa }_i$ is a principal curvature. Consequently regular interrogation fails at $d{\kappa }_i=1$; on regular pieces the offset principal curvatures are ${\kappa }_i/(1-d{\kappa }_i)$ with the continued normal orientation. Global self-intersections can occur even when both local factors are nonzero. Evaluate and differentiate the base surface and its normalized normal; an exact conversion back to the original patch type is unnecessary.