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Parallel Body

A parallel body, or outer parallel body, of a convex body K subset= R^n at distance r>=0 is the Minkowski sum

K_r=K+rB^n={x in R^n:d(x,K)<=r},

where B^n is the closed ball of radius 1 centered at the origin, rB^n={rb:b in B^n}, and d(x,K)=inf_(y in K)||x-y|| uses the Euclidean metric. The same construction for any nonempty compact set K is called an outer parallel set (Gardner 2006, p. 9).

For example, the parallel body of the square Q=[-1,1]^2 is

Q_r={(x,y):max(|x|-1,0)^2+max(|y|-1,0)^2<=r^2}.

For r>0, its boundary consists of four line segments joined by four quarter-circles of radius r (Trott 2006). Its area is 4+8r+pir^2, obtained by adding four rectangles and four quarter-disks to the original square. For a convex body in the plane with a smooth curve as its boundary, the boundary of K_r is the outer parallel curve at distance r.

See also

Closed Ball, Convex Body, Minkowski Sum, Parallel Curves

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References

Gardner, R. J. Geometric Tomography, 2nd ed. Cambridge, England: Cambridge University Press, p. 9, 2006. https://doi.org/10.1017/CBO9781107341029.Pottmann, H. and Hofer, M. "Geometry of the Squared Distance Function to Curves and Surfaces." In Visualization and Mathematics III (Ed. H.-C. Hege and K. Polthier). Berlin, Germany: Springer-Verlag, pp. 221-242, 2003. https://doi.org/10.1007/978-3-662-05105-4_12.Sturm, T. "An Algebraic Approach to Offsetting and Blending of Solids." In Computer Algebra in Scientific Computing: CASC 2000 (Ed. V. G. Ganzha, E. W. Mayr, and E. V. Vorozhtsov). Berlin, Germany: Springer-Verlag, pp. 367-381, 2000. https://doi.org/10.1007/978-3-642-57201-2_28.Sturm, T. and Weispfenning, V. "An Algebraic Approach to Offsetting and Blending of Solids." Technical Report MIP-9804. Passau, Germany: Fakultät für Mathematik und Informatik, Universität Passau, 1998. https://webdoc.sub.gwdg.de/ebook/e/2001/mip/mip-berichte.html.Trott, M. "Polynomials in Inequalities." §1.2.3 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28815-5. https://www.mathematicaguidebooks.org/.

Cite this as:

Weisstein, Eric W. "Parallel Body." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ParallelBody.html

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