Clifford algebra, Lorentzian geometry, and rational parametrization of canal surfaces

We present a new approach toward the rational parametrization of canal surfaces. According to our previous work, every canal surface with rational (respectively polynomial) spine curve and rational (respectively polynomial) radius function is a rational (respectively polynomial) Pythagorean hodograph curve in R 3,1 . Drawing upon this formalism and utilizing the underlying Lorentzian geometry, the problem is reduced to simple algebraic manipulations. We also illustrate how our work relates to the previous work of Pottmann and Peternell. Finally, we give an outline of an approach toward the rotation-minimizing parametrization of canal surfaces.