Cubic Trigonometric Nonuniform Spline Curves and Surfaces

Abstract

A class of cubic trigonometric nonuniform spline basis functions with a local shape parameter is constructed. Their totally positive property is proved. The associated spline curves inherit most properties of usual polynomial B-spline curves and enjoy some other advantageous properties for engineering design. They have C2 continuity at single knots. For equidistant knots, they have C3 continuity and C5 continuity for particular choice of shape parameter. They can express freeform curves as well as ellipses. The associated spline surfaces can exactly represent the surfaces of revolution. Thus the curve and surface representation scheme unifies the representation of freeform shape and some analytical shapes, which is popular in engineering.