The approximation of non-degenerate offset surfaces

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Abstract

This paper describes algorithms for approximating the offsets to general piecewise parametric surfaces by networks of bicubic patches. The surface to be offset may be composed of arbitrary parametric patches, provided precise tangent continuity obtains across all patch boundaries and the surface metric is everywhere non-singular (i.e., a unique surface normal is defined).

The approximation scheme consists of three stages: (1) A differential surface analysis is performed to ascertain the extremum principal radii of curvature for each patch — this constrains the offset magnitude that may be specified without degeneration of the offset. (2) The parametric domain of each patch is sub-divided, and a bicubic approximant to the offset for each sub-domain is computed. (3) A tolerance analysis is performed on each patch of the offset approximation to evaluate its extremum deviations from the true offset.

The accuracy of the offset approximation increases with the degree of subdivision of the parametric domain of each patch. Fractional errors in the range 10−2 to 10−3 are typical for two- or three- fold sub-division of each parametric variable (four- and nine- fold area division respectively), provided the surface is smooth and its smallest concave radius of curvature considerably exceeds the specified offset magnitude. A fully automatic tolerance-based surface offset capability may be developed by providing feedback between stages (3) and (2), successive degrees of parametric sub-division being determined by the errors from prior approximations until the desired accuracy is achieved.

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