Interpolating curves with gradual changes in curvature (Q1099574)

The problem of interpolating a set of points by a parametric curve \({\mathcal R}(t)=[x(t),y(t),z(t)]\) is investigated. The behavior of norms \({\mathcal J}_ i=\int^{1}_{0}| {\mathcal R}^{(i)}(t)|^ 2dt,\) where \(i=2,3,4\) and \({\mathcal R}^{(i)}(t)\) denotes the i-th derivative of \({\mathcal R}(t)\) is studied. The problem of generating a curve that minimizes the norm \({\mathcal J}_ 2\) can be solved by cubic spline interpolation, and geometric splines [\textit{H. Hagen}, Computer Aided Geometric Design 2, 223-227 (1985; Zbl 0577.65006)] can be used to minimize the norm \({\mathcal J}_ 3.\) In the paper under review, a different approach based on a continuous polynomial \({\mathcal R}(t)\) of sufficiently high degree to meet all interpolating conditions and to have enough extra freedoms to reduce one of norms \({\mathcal J}_ 2\), \({\mathcal J}_ 3\) or \({\mathcal J}_ 4\) to a practically acceptable minimum is outlined. Some conclusions and examples are presented, too.