Visualization and analysis of regions of monotonic curvature for interpolating segments of extended sectrices of Maclaurin

摘要

In biangular coordinates, a G~1 Hermite interpolation reduces to the problem of choosing appropriate functions interpolating the coordinates of the curve at its endpoints. The simplest linear equations in biangular coordinates correspond to a sectrix of Maclaurin, which can be extended by introducing two shape parameters that stretch the curve toward the sides of its triangular envelope. However, these additional degrees of freedom complicate the possibility of obtaining analytical conditions for curvature monotonicity. Therefore, we determine the regions of monotonic curvature experimentally, for different values of the interpolant's shape parameters. We show that the parameter space of curves based on a sectrix of Maclaurin contain larger regions of monotonic curvature than those of a quadratic Bézier curve and its rational form. This suggests that our approach provides more flexibility in designing applications requiring curvature monotonicity. Finally, we classify curves with monotonic curvature based on their regions of monotonic curvature.