A Class of Quasi-Quartic Trigonometric BÉZier Curves and Surfaces

Abstract

A new kind of quasi-quartic trigonometric polynomial base functions with a shape parameter \u03bb over the space \u2126=span {1, sint, cost, sint2t, cos2t} is presented, and the corresponding quasi-quartic trigonometric B\u00e9zier curves and surfaces are defined by the introduced base functions. The quasi-quartic trigonometric B\u00e9zier curves inherit most of properties similar to those of quartic B\u00e9zier curves, and can be adjusted easily by using the shape parameter \u03bb. The jointing conditions of two pieces of curves with G2 and C4 continuity are discussed. With the shape parameter chosen properly, the defined curves can express exactly any plane curves or space curves defined by parametric equation based on{1, sint, cost, sint2t, cos2t} and circular helix with high degree of accuracy without using rational form. The corresponding tensor product surfaces can also represent precisely some quadratic surfaces, such as sphere, paraboloid, cylindrical surfaces, and some complex surfaces. The relationship between quasi-quartic trigonometric B\u00e9zier curves and quartic B\u00e9zier curves is also discussed, the larger is parameter \u03bb, and the more approach is the quasi- quartic trigonometric B\u00e9zier curve to the control polygon. Examples are given to illustrate that the curves and surfaces can be used as an efficient new model for geometric design in the fields of CAGD.