Rational parametrization of canal surface by 4 dimensional Minkowski Pythagorean hodograph curves

Lorentzian geometry with a Minkowski Pythagorean hodograph (MPH) formalism in ℝ^3,1 gives us a new and insightful method of rational parametrization of canal surfaces. Our previous works about MPH curves in ℝ^3,1 shows that a curve γ(t)=(x(t),y(t),z(t),r(t)) in ℝ^3,1 can be represented by the PH representation map in script Cℓ(3,1) which avoids the complex root finding algorithm. Our parametrization method gives us the flexibility to represent the canal surfaces within their fiber ambiguities. This paper constitutes the first step of our ongoing work which deals with the issues for canal surfaces in a truly new and intriguing manner such as finding rotation minimizing frames. We believe this is just the tip of the iceberg and the further work will yield many valuable applications in the area of canal surfaces.