= Solution
A <parametric surface interrogation> interface must specify its parameter domain, patch boundaries, trimming restrictions and any exceptional or singular points. It must evaluate $S(u,v)$ and the available <derivatives>, in particular $S_u,S_v$ for <tangent vectors> and
$$
n=\frac{S_u\times S_v}{\|S_u\times S_v\|}
$$
for the oriented <normal vector> where the denominator is nonzero. Second derivatives $S_{uu},S_{uv},S_{vv}$ are needed when the code requests curvature or second-order geometric error estimates; higher derivatives depend on the enquiry. The interface must report where these quantities exist, rather than assign an artificial normal at a singular parameter point.
Global searches also need subdivision into equivalent restricted patches, certified <bounding volumes> and a geometric approximation/error bound. These support intersection isolation, pruning and tolerance-controlled meshing. A single point-evaluation routine cannot guarantee a global search because it may miss remote branches or features between samples. Adjacent-patch topology and consistent orientation are needed to join results and interpret the represented surface.
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