Cubic Trigonometric Hermite Interpolation Curve: Construction, Properties, and Shape Optimization

Abstract

Cubic Hermite interpolation curve plays a very important role in interpolation curves modeling, but it has three shortcomings including low continuity, difficult shape adjustment, and the inability to accurately represent some common engineering curves. We construct a cubic trigonometric Hermite interpolation curve to make up the three shortcomings of cubic Hermite interpolation curve once and for all. The cubic trigonometric Hermite interpolation curve not only inherits the features of cubic Hermite interpolation curve but also achieves C2 continuity, has local and global adjustability, and can accurately represent elliptical arc, circular arc, quadratic parabolic arc, cubic parabolic arc, and astroid arc that often appear in engineering. In addition, we give the schemes for optimizing the shape of the cubic trigonometric Hermite interpolation curve based on internal energy minimization. The schemes include optimizing the shape of planar curve and spatial curve. Some modeling examples show that the proposed schemes are effective and the cubic trigonometric Hermite interpolation curve is more practical than cubic Hermite interpolation curve.

Parametric Curve Comparison for Modeling Floating Offshore Wind Turbine Substructures

“…The CHS is commonly used to construct interpolation curves in engineering modelling and designs for providing solutions to practical engineering problems [22,23].…”

A quartic trigonometric interpolatory spline with local free parameters

“…However, trigonometric functions have also received very much attention within geometric modeling. Such as the trigonometric Be´zier curves, [1][2][3][4] the trigonometric B-spline curves, [5][6][7] the trigonometric Hermite curves, 8,9 the trigonometric rational curves, 10,11 etc. The curves constructed in trigonometric function space have been applied in signal analysis, 12 kitchen product design, 13 robot path planning, 14 and other engineering problems.…”