Tensor-product surface patches with Pythagorean-hodograph isoparametric curves

The construction of tensor-product surface patches with a family of Pythagorean-hodograph (PH) isoparametric curves is investigated. The simplest nontrivial instances, interpolating four prescribed patch boundary curves, involve degree (5,4) tensor-product surface patches x(u,v) whose v=constant isoparametric curves are all spatial PH quintics. It is shown that the construction can be reduced to solving a novel type of quadratic quaternion equation, in which the quaternion unknown and its conjugate exhibit left and right coefficients, while the quadratic term has a coefficient interposed between them. A closed-form solution for this type of equation is derived, and conditions for the existence of solutions are identified. The surfaces incorporate three residual scalar freedoms which can be exploited to improve the interior shape of the patch. The implementation of the method is illustrated through a selection of computed examples.