Volumes with piecewise quadratic medial surface transforms: Computation of boundaries and trimmed offsets

MOS surfaces are rational surfaces in R which provide rational envelopes of the associated two-parameter family of spheres. Moreover, all the offsets admit rational parameterizations as well. Recently, it has been proved that quadratic triangular Bézier patches in R are MOS surfaces. Following this result, we describe an algorithm for computing an exact rational envelope of a two-parameter family of spheres given by a quadratic patch in R. The main focus of this paper is given to geometric aspects of the algorithm. Since these patches are capable of producing C smooth approximations of medial surface transforms of spatial domains, we use this algorithm to generate rational approximations of envelopes of general medial surface transforms. One of the main advantages of this approach to offsetting is the fact that the trimming procedure becomes considerably simpler.