CSG set-theoretic solid modelling and NC machining of blend surfaces
SCG '86 Proceedings of the second annual symposium on Computational geometry
Blending quadric surfaces with quadric and cubic surfaces
SCG '87 Proceedings of the third annual symposium on Computational geometry
An extension of the potential method to higher-order blendings
SMA '91 Proceedings of the first ACM symposium on Solid modeling foundations and CAD/CAM applications
Proceedings of the 18th annual conference on Computer graphics and interactive techniques
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We survey the potential method for blending implicit algebraic surfaces, summarizing and extending work previously reported. The method is capable of deriving blends for pairs of algebraic surfaces, and is guaranteed to produce blending surfaces of lowest possible degree for two quadrics in general position. We give two paradigms by which to understand the method. The first paradigm views the blends as surfaces swept out by a family of space curves. The second, more general paradigm considers the surfaces as result of deformation of a parameter space effected by substitution. The method has a general formulation based on projective parameter spaces, but is also the image under projective transformation of the simpler, affine formulation. The deformation by substitution paradigm is extended to blend blending surfaces at solid vertices without a degree penalty, under the assumption that the vertex valence has been reduced to three. It may also lead to a general solution for blending patches of algebraic surfaces that meet tangentially. A special case of this problem is solved and illustrated.