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Offset and Differential Geometry in Mathematica

Marek Byrtus
Organization: University of West Bohemia
Department: Department of Mathematics
Faculty of Applied Sciences

2007 Wolfram Technology Conference
Conference location

Champaign, IL


The theory of the offsets (sometimes and in special cases called the equidistante or the envelope hypersurface) is a very useful mathematical approach, which is widely used in machine cutting. The offset of the given plane curve or surface describes the motion of the cutter. The aim of machine cutting is to find problematic parts of the curve or the surface where could arise the crucial problem - the undercut. One way to identify the undercut problem is to study the offset of a given object. Another approach is to study the curvatures of a given object and the cutter in a corresponding point, which belongs to the part of differential geometry.

Mathematica does not provide the functions to compute the offset of a given object and also the functions from differential geometry like curvatures, etc. This was the reason to develop this part of a function in Mathematica. There were used very powerful symbolic computations in Mathematica to create a new package where are the offsets and differential geometry mainly for polynomial and rational parametrizations of the curves and the surfaces.

The talk will present this new package, which consists the following items:
  • Offset (two- and three-dimensional, reparametrization)
  • Differential Geometry (curvatures, fundamental forms of surfaces, Dupin Indicatrix)
  • Conic Section (discussion of conic section, their useful properties)
  • Part of Algorithms for solving the undercut problem (how to indicate the undercut)

*Mathematics > Geometry
*Wolfram Technology


OffsetAndDifferentialGeometry.zip (1.1 MB) - ZIP archive