Rational adaptive blends among obstacles in 3D by contour method

In this paper we will continue in investigating 'contour method' and its using for the computation of rational parameterizations of canal surfaces without a need of sum of squares (SOS) decomposition. Further approaches for constructing flexible smooth transitions between canal surfaces will be presented. Mainly, we focus on one particular application of recently introduced rational envelope curves, newly constructed over an arbitrary planar rational curve in space. Using this type of curves significantly simplifies the previous methods discussed in Bizzarri (2015), and mainly new situations, which could not have been handled with the previous setup, are successfully solved, now. Especially a method for constructing rational adaptive blends which bypass a given obstacle (or more given obstacles when needed) is thoroughly discussed and its functionality is demonstrated on a number of examples. The designed approach works not only for simple obstacles represented by one-dimensional medial axis transforms but also for more general obstacles described by two-dimensional medial surface transforms.