Tensor-product surface patches with Pythagorean-hodograph isoparametric curves

Abstract

The construction of tensor-product surface patches with a family of Pythagorean-hodograph (PH) isoparametric curves is investigated. The simplest non-trivial 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 closedform 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.

Solution of a quadratic quaternion equation with mixed coefficients

“…Finally, substituting these expressions into the third equation yields the form (1), where P, Q, R, S are known, and we set X = A 11 . Complete details may be found in [8].…”

“…Finally, substituting these expressions into the third equation yields the form (1), where P, Q, R, S are known, and we set X = A 11 . Complete details may be found in [8]. 4 The solution procedure Different approaches to equation ( 1) are appropriate, according to whether or not the coefficient P lies on the real axis R. These cases will be analyzed in detail in Sections 4.1 and 4.2 below.…”

“…in this case, the set of solutions in H is the circle Further examples of numerical solutions to equation (1), computed in the context of the surface construction problem (Section 3), are presented in [8].…”

“…In the present study, we have considered a special quadratic quaternion equation in the quaternion variable and its conjugate, with mixed coefficients. This equation, arising from a surface construction problem [8], was shown to admit a complete characterization of its solutions, for all possible instances of the coefficients. In addition to point solutions, circles or 3-spheres of solutions are observed -as distinct from the case of unilateral coefficients, which admits [10,19,23] only point solutions and 2-spheres of solutions.…”

“…The motivation for studying equation (1) arises [8] from the construction of a surface patch x(u, v) defined on (u, v) ∈ [ 0, 1 ] 2 with prescribed boundary curves, such that the v = constant isoparametric curves are all polynomial Pythagorean-hodograph (PH) curves [5]. A brief review of the surface construction problem can be found in Section 3 below.…”