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Our method is based on B\u00e9zier clipping, which is a bracketing\u2010based root finding method that is commonly\u2010used in computer\u2010aided geometric design. Compared to B\u00e9zier bisection, our clipping method exhibits significantly faster convergence and reduces duplicate solutions. Compared to the general Newton\u2010Raphson algorithm, the success of our approach does not depend on the quality of an initial guess. The B\u00e9zier\u2010based formulation readily generalizes to polynomial higher\u2010order interpolants, such as for tricubic interpolation. With this, we derive the acceleration in the Sujudi\u2010Haimes vortex coreline criterion directly from the vector field interpolant. In addition, we describe the cross product of two polynomial vector fields in arbitrary polynomial degree, which can be used to approximate non\u2010polynomial interpolants. Further, we examine the effect of Newton refinement on the proposed B\u00e9zier clipping method. 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