Symbolic parametrization of pipe and canal surfaces

A canal surface <italic>S</italic>, generated by a parametrized curve <italic>m</italic>(<italic>t</italic>), in R<supscrpt>3</supscrpt> is the envelope of the set of spheres with radius <italic>r</italic>(<italic>t</italic>) centered at <italic>m</italic>(<italic>t</italic>). This concept generalizes the classical offsets (for <italic>r</italic>(<italic>t</italic>) = <italic>const</italic>) of plane curves. In this paper we develop elementary symbolic methods for generating a rational parametrization of canal surfaces generated by rational curves <italic>m</italic>(<italic>t</italic>) with rational radius variation <italic>r</italic>(<italic>t</italic>). In a pipe surface <italic>r</italic>(<italic>t</italic>) is constant.

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