A Class of Quasi-Quartic Trigonometric BÉZier Curves and Surfaces

A new kind of quasi-quartic trigonometric polynomial base functions with a shape parameter λ over the space Ω=span {1, sint, cost, sint2t, cos2t} is presented, and the corresponding quasi-quartic trigonometric Bézier curves and surfaces are defined by the introduced base functions. The quasi-quartic trigonometric Bézier curves inherit most of properties similar to those of quartic Bézier curves, and can be adjusted easily by using the shape parameter λ. The jointing conditions of two pieces of curves with G and C 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 Bézier curves and quartic Bézier curves is also discussed, the larger is parameter λ, and the more approach is the quasiquartic trigonometric Bézier 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.