Blending multiple parametric normal ringed surfaces using implicit functional splines and auxiliary spheres

First presented by Hartmann, closings (implicit surfaces sealing the inlets or outlets of pipes) can bridge the gap between parametric pipe surfaces and implicit functional splines (a powerful tool for blending several implicit surfaces). This paper proposes auxiliary spheres instead of the initial pipe surfaces as the base surfaces in constructing closings, so that the closing based algorithm of two steps (constructing a closing for each pipe and blending the closings) can G1-continuously connect multiple parametric normal ringed surfaces with freeform directrices and variable radii. The basic theory of an auxiliary sphere tangent to the normal ringed surface is addressed. Either one or two (yielding more design parameters) auxiliary spheres can be added. How the parameters influence the closing configuration is discussed. In addition, the blending shape can be optimized by genetic algorithm after assigning some fiducial points on the blend. The enhanced algorithm is illustrated with four practical examples.