Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 77 / 4 / i

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.