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 G^1-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.

[1]  Erich Hartmann,et al.  Parametric Gn blending of curves and surfaces , 2001, The Visual Computer.

[2]  John E. Hopcroft,et al.  Automatic surface generation in computer aided design , 2005, The Visual Computer.

[3]  Renhong Wang,et al.  Functional splines with different degrees of smoothness and their applications , 2008, Comput. Aided Des..

[4]  E. Hartmann.  Blending of implicit surfaces with functional splines , 1990, Comput. Aided Des..

[5]  Erich Hartmann.  The normalform of a space curve and its application to surface design , 2001, Vis. Comput..

[6]  Rimvydas Krasauskas,et al.  Branching blend of natural quadrics based on surfaces with rational offsets , 2008, Comput. Aided Geom. Des..

[7]  Josef Hoschek,et al.  Gn∗-functional splines for interpolation and approximation of curves, surfaces and solids , 1990, Comput. Aided Geom. Des..

[8]  M. J. Pratt,et al.  Cyclides in computer aided geometric design , 1990, Comput. Aided Geom. Des..

[9]  Ching-Kuang Shene.  Blending two cones with Dupin cyclides , 1998, Comput. Aided Geom. Des..

[10]  James H. Anderson,et al.  Constructive modeling of G1 bifurcation , 2002, Comput. Aided Geom. Des..

[11]  Seth Allen,et al.  Results on nonsingular, cyclide transition surfaces , 1998, Comput. Aided Geom. Des..

[12]  Michael J. Pratt.  Cyclides in computer aided geometric design II , 1995, Comput. Aided Geom. Des..

[13]  Panagiotis D. Kaklis,et al.  G1-smooth branching surface construction from cross sections , 2007, Comput. Aided Des..

[14]  Joe D. Warren,et al.  Blending algebraic surfaces , 1989, TOGS.

[15]  L. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communications.

[16]  Yun-shi Zhou,et al.  On blending of several quadratic algebraic surfaces , 2000, Comput. Aided Geom. Des..

[17]  Nicholas M. Patrikalakis,et al.  Analysis and applications of pipe surfaces , 1998, Comput. Aided Geom. Des..

[18]  Erich Hartmann,et al.  Gn-continuous connections between normal ringed surfaces , 2001, Comput. Aided Geom. Des..

[19]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[20]  Insung Ihm,et al.  Algebraic surface design with Hermite interpolation , 1992, TOGS.

[21]  Falai Chen,et al.  Blending Quadric Surfaces with Piecewise Algebraic Surfaces , 2001, Graph. Model..

[22]  G. Farin.  Curves and Surfaces for Cagd: A Practical Guide , 2001 .

[23]  Helmut Pottmann,et al.  Computing Rational Parametrizations of Canal Surfaces , 1997, J. Symb. Comput..

[24]  Günter Aumann,et al.  Curvature continuous connections of cones and cylinders , 1995, Comput. Aided Des..