In this paper a $G^2$ continuous geometric interpolation of Hermite data by Pythagorean-hodograph (PH) quintic curves in $\RR^d$ is considered. For two sets of appropriate Hermite data (tangent directions and curvature vectors) at two distinct points a PH quintic which interpolates given data geometrically is sought. The problem reduces to solving a system of nonlinear algebraic equations involving only geometric interpolation parameters as unknowns. Several solutions of the same quality (considering the shape of the resulting interpolants) might exist and a thorough asymptotic analysis is done to establish the existence of an odd number of asymptotic solutions having the best approximation order, i.e., order six in this case. Such a solution for a particular set of data is then traced by homotopy to find an appropriate solution for general data. Numerical examples confirm that the method is efficient in practical computations.
The problem of geometric interpolation by Pythagorean-hodograph (PH) curves of general degree ▫$n$▫ is studied independently of the dimension ▫$d \ge 2$▫. In contrast to classical approaches, where special structures that depend on the dimension are considered (complex numbers, quaternions, etc.), the basic algebraic definition of a PH property together with geometric interpolation conditions is used. The analysis of the resulting system of nonlinear equations exploits techniques such as the cylindrical algebraic decomposition and relies heavily on a computer algebra system. The nonlinear equations are written entirely in terms of geometric data parameters and are independent of the dimension. The analysis of the boundary regions, construction of solutions for particular data and homotopy theory are used to establish the existence and (in some cases) the number of admissible solutions. The general approach is applied to the cubic Hermite and Lagrange type of interpolation. Some known results are extended and numerical examples provided.
COBISS.SI-ID: 16051289
In the paper, the Lagrange geometric interpolation by spatial rational cubic Bezier curves is studied. It is shown that under some natural conditions the solution of the interpolation problem exists and is unique. Furthermore, it is given in a simple closed form which makes it attractive for practical applications. Asymptotic analysis confirms the expected approximation order, \ie, order six. Numerical examples pave the way for a promising nonlinear geometric subdivision scheme.
COBISS.SI-ID: 16207449