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A parallel surface is a constant-distance normal displacement $P_d=P+dn$ of a regular surface with unit normal vector $n$. With shape operator $W=-Dn$, its differential is $(I-dW)DP$; it can become singular where $1-d{\kappa }_i=0$ for a principal curvature ${\kappa }_i$.

// Target: geometry-and-topology.bigb

A parallel surface at signed distance $d$ from a design surface describes a constant-thickness skin, shell, coating or clearance envelope. A particularly important application is machining with a spherical cutter of radius $r$: when the cutter touches $P$ without penetration, its center lies at $P+rn$, where $n$ is the outward unit normal vector. Thus a tool-center path can be computed on the surface offset rather than on the desired contact surface. Positive and negative surface offsets also model material addition and removal, and pairs of offsets describe the two faces of a thin manufactured wall. Offsets convert contact, thickness and clearance specifications into geometric surfaces which can be interrogated.

The construction is initially local. Where the offset develops self-intersections or reaches a curvature radius, not every point of the raw parallel surface is a usable physical boundary. Tool paths and solid envelopes may require trimming those pieces and checking global collisions.

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.

A variable normal offset displaces a parametric surface along its unit normal vector by a scalar field $d(u,v)$. Unlike a constant-distance parallel surface, its derivatives also contain $d_{u}n$ and $d_{v}n$.

// Target: analysis.bigb