Paralleelipinnats

Paralleelipinnats, in English parallel surfaces, refers to a family of surfaces in three-dimensional space formed by offsetting a given smooth surface by a fixed distance along its unit normal vector. For a surface S with unit normal n(p) at each point p, the parallel surface at distance d is the set S_d = { p + d n(p) : p ∈ S }. This construction yields a new surface that runs at a constant normal distance from the original one.

The offset is well-defined where the normal map is regular; at points where the curvature becomes singular (the focal set), the parallel surface can develop singularities if the distance d reaches certain critical values. If κ1 and κ2 are the principal curvatures of S, the principal curvatures of the parallel surface S_d are κ1^d = κ1 / (1 + d κ1) and κ2^d = κ2 / (1 + d κ2), provided 1 + d κi ≠ 0. Consequently, parallel surfaces preserve the general shape while adjusting curvature and other geometric measures in a distance-dependent way. For small |d|, the offset remains regular and smoothly related to the original surface.

Common examples illustrate the concept: offsetting a plane yields another plane parallel to it; offsetting a sphere by distance d produces a concentric sphere with radius increased by d; offsetting a cylinder yields a parallel cylindrical surface coaxial with the original.

Applications of paralleelipinnats include computer-aided design and manufacturing (tool paths and allowances), computer graphics (shadowing and rendering), architectural design, and medical imaging (offsetting anatomical surfaces for visualization and analysis).

See also: offset curves and surfaces, Gauss map, normal congruences.