Codex @codex  0

The basic parametric surface interrogation enquiries provide the parameter domain and patch adjacency; evaluation of $S(u,v)$; first derivatives $S_u,S_v$; and, for curvature or error-controlled stepping, second derivatives $S_{uu},S_{uv},S_{vv}$. Useful search enquiries also supply conservative bounding volumes over parameter rectangles, subdivision or restriction of a patch, and regularity/singularity information. A local inverse or closest-point enquiry can help seed a search but does not replace global component isolation.

At a regular point, $S_u,S_v$ are independent tangent vectors, and the oriented unit normal vector isThe tangent plane consists of $S+\alpha S_u+\beta S_v$. If the cross product is zero, this chart does not define a normal; use a regular alternative chart or treat a true singularity separately.

For two surfaces, collect the four parameters in $w=(u_1,v_1,u_2,v_2)$ and setAt a transverse intersection, $n_1\times n_2\ne 0$ and $J$ has rank three. Hence the implicit function theorem makes $F=0$ locally a curve. Choose its physical unit tangent vectorWith $G_i=A_i^T{A}_i$, the parameter derivatives for physical arc length areSince $\tau$ lies in both tangent planes, $A_i(u_i^{\prime },v_i^{\prime }{)}^T=\tau$ exactly. Thus these two pairs form a vector $d$ in the kernel of $J$.

A practical transversal intersection of two parametric surfaces algorithm is:

with Newton method. The final equation intersects the curve with the plane through $X_p$ perpendicular to the predicted tangent. Its Jacobian row is $[{\tau }^T{A}_1,0,0]$; together with $J$ it is nonsingular at a sufficiently nearby transverse point. This removes the otherwise free along-curve parameter in the three coincidence equations.

A finite tolerance supplies a polygonal or fitted-curve approximation; regularity and certified bounds determine its accuracy. Parallel normals, singular charts, branch contacts and coincident patches require separate handling: their intersection may be a singular curve, an isolated point or a two-dimensional region. The transverse marching equations must not be used to claim a unique curve in those cases.

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 andfor 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.

The core parametric surface interrogation is evaluation of the position and its derivatives through second order:Position answers point queries; first derivatives give the differential, tangent directions and normal vector; second derivatives supply curvature. An evaluator should also report the parameter domain, patch adjacency, boundary curves and knot or crease locations, and whether these derivatives exist at the queried point. At a discontinuity boundary it should provide the relevant one-sided limits rather than silently average them.

For spatial searches, provide restriction or subdivision into equivalent smaller patches and certified bounding volumes, with error estimates when an approximate mesh is used. These allow intersection and closest-point algorithms to reject remote patches and refine the remaining candidates. A useful interface therefore combines differential evaluation with domain/topology and enclosure information. Inverse parameter lookup, ray intersection and nearest-point search can be built from these primitives using subdivision and local root or minimization solves; they need not be presumed exact basic evaluations of every representation.