A Shape Preserving Class of Two-Frequency Trigonometric B-Spline Curves

This paper proposes a new approach to define two frequency trigonometric spline curves with interesting shape preserving properties. This construction requires the normalized B-basis of the space U4(I alpha)=span{1,cost,sint,cos2t,sin2t} defined on compact intervals I alpha=[0,alpha], where alpha is a global shape parameter. It will be shown that the normalized B-basis can be regarded as the equivalent in the trigonometric space U4(I alpha) to the Bernstein polynomial basis and shares its well-known symmetry properties. In fact, the normalized B-basis functions converge to the Bernstein polynomials as alpha -> 0. As a consequence, the convergence of the obtained piecewise trigonometric curves to uniform quartic B-Spline curves will be also shown. The proposed trigonometric spline curves can be used for CAM design, trajectory-generation, data fitting on the sphere and even to define new algebraic-trigonometric Pythagorean-Hodograph curves and their piecewise counterparts allowing the resolution of C(3 Hermite interpolation problems.