Trigonometric Polynomial Curve and Surface with many Advantages

To make the single curve and surface model possesses shape adjustability, high order continuity, automatic interpolation, and exact representation of conic curve and surface, a set of quintic trigonometric polynomial blending function with two parameters is constructed, and its properties are discussed. Based on the function, a curve on four-point piecewise and a surface on sixteen-point piecewise are defined, and their properties are given. The piecewise combination structure makes the curve and surface have local control property. The effect of changing the parameter on the shape of the curve is discussed. We prove that when taking special parameters, the curve can achieve G⁵ or G⁷ continuity and it is still shape adjustable when it is G⁵ continuous. When one of the two parameters is equal to a fixed value, the new curve and surface can interpolate the given set of points without computing the control points. The interpolation curve and surface are also shape adjustable. The condition for the curve and surface represent the ellipse and ellipsoid is given. Experimental results show the correctness and effectiveness of the presented curve and surface representation method.