The paper presents an interpolation scheme for $G^1$ Hermite motion data, i.e., interpolation of data points and rotations at the points, with spatial quintic Pythagorean-hodograph curves so that the Euler-Rodrigues frame of the curve coincides with the rotations at the points. The interpolant is expressed in a closed form with three free parameters, which are computed based on minimizing the rotations of the normal plane vectors around the tangent and on controling the length of the curve. The proposed choice of parameters is supported with the asymptotic analysis. The approximation error is of order four and the Euler-Rodrigues frame differs from the ideal rotation minimizing frame with the order three. The scheme is used for rigid body motions and swept surface construction.
COBISS.SI-ID: 1024447572
Polynomial geometric interpolation by parametric curves has become 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 G^2 and quartic G^3 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.
COBISS.SI-ID: 16449369
In this paper the C^1 Hermite interpolation problem by spatial Pythagorean-hodograph cubic biarcs is presented and a general algorithm to construct such interpolants is described. Each PH cubic segment interpolates C^1 data at one point and they are then joined together with a C^1 continuity at some unknown common point sharing some unknown tangent vector. Biarcs are expressed in a closed form with three shape parameters. Two of them are selected based on asymptotic approximation order, while the remaining one can be computed by minimizing the length of the biarc or by minimizing the elastic blending energy. The final interpolating spline curve is globally C^1 continuous, it can be constructed locally and it exists for arbitrary Hermite data configurations.