Hermite Geometric Interpolation by Rational Bézier Spatial Curves

Polynomial geometric interpolation by parametric curves became one of the standard techniques for interpolation of geometric data. An obvious generalization leads to rational geometric interpolation schemes, which are a much less investigated research topic. The aim of this paper is to present a general framework for Hermite geometric interpolation by rational Bézier spatial curves. In particular, cubic G2 and quartic G3 interpolations are analyzed in detail. Systems of nonlinear equations are derived in a simplified form and the existence of admissible solutions is studied. For the cubic case, geometric conditions implying solvability of the nonlinear system are also stated. The asymptotic analysis is done in both cases and optimal approximation orders are proved. Numerical examples are given, which confirm the theoretical results.