Paralleelset

Paralleelset is a geometric construction associated with a subset A of Euclidean space. For a nonnegative radius r, the r-parallel set of A, denoted A_r, is the set of all points whose distance to A is at most r. Equivalently, A_r equals the Minkowski sum of A with the closed ball B_r of radius r: A_r = A ⊕ B_r.

As r increases, A_r expands and grows continuously with r; if A is bounded, A_r remains bounded for finite r. If A is closed, A_r is closed; A_0 = A. The boundary ∂A_r is called the offset of ∂A, particularly when A has a sufficiently regular boundary. For smooth A, ∂A_r is obtained by offsetting ∂A outward by distance r.

In the plane, and for convex A, the area of A_r satisfies the Steiner formula: Area(A_r) = Area(A) + Perimeter(A) · r + π · r^2. More generally in R^n, Vol_n(A_r) can be expressed as a polynomial in r with coefficients depending on A (intrinsic volumes or quermassintegrals). Similar considerations apply to the surface area and other measures.

Variants include the inner parallel set A ⊖ B_r = { x in A : distance(x, A^c) ≥ r }, the set of points at least distance r from the complement. Paralleelset and related constructions are used to study reach, curvature, and regularity of sets.

Applications of parallel sets appear across disciplines, including computer-aided design, geographic information systems, and image processing (where dilation with a ball operates as a structuring element). They provide a framework for buffers, offsets, and the analysis of boundary geometry.