A Class of Algebraic Trigonometric Interpolation Splines and Applications

A new kind of interpolation splines with a shape parameter over the algebraic trigonometric function space Ω=span {1, t, sint, cost, sint2t, cos2t} is presented, which are called cubic algebraic trigonometric splines. The cubic algebraic trigonometric splines have many similar properties to cubic B-splines. The corresponding spline curves and surfaces can interpolate directly some control points without solving system of equations or inserting some additional control points. The curves can be used to represent exactly straight line segment, circular arc, elliptic arc, parabola and some transcendental curves such as circular helix. The corresponding tensor product surfaces can also represent precisely some quadratic surfaces and transcendental surfaces, such as sphere, cylindrical surfaces, and some complex surfaces such as helix tube can be constructed by these basic surfaces exactly. The shape of the curves and surfaces can be modified globally through changing the values of the parameters. Moreover, these curves and surfaces are C² continuous when choosing proper shape parameters. Examples showed that the cubic algebraic trigonometric interpolation splines can be used as an efficient new model for geometric design in the fields of CAGD.