C cubic trigonometric B-spline curves with a real parameter

This paper presents a class of cubic trigonometric B-spline curves with a real parameter , which are called -B-spline curves. The -B-spline curves have properties similar to those of the standard cubic uniform B-spline curves. In particular, the -B-spline curves are 3 and can achieve shape adjustment by altering the value of the parameter even if the control points are kept unchanged. With proper conditions, the -B-spline curves also can exactly represent arcs of ellipses and parabolas. In order to obtain smooth -B-spline curves, a method for determining the value of parameter is presented.


INTRODUCTION
In geometric modelling, it is often necessary to freely change the shapes of curves. Hence, the curves with parameters have been paid more and more attention by  (Liu et al., 2010). Those curves with parameters inherit the similar to or the same properties as the corresponding standard curves, but also have better performance ability because of the parameters. The shapes of these curves can be adjusted by altering the values of the parameters even if the control points the corresponding standard curves. Moreover, some of these curves can exactly represent ellipses and parabolas without complicated conditions. Therefore, the curves with parameters have wide applications in engineering.
It is known that the B-spline technique is one of the methods of analytic representation of curves and surfaces that have won wide acceptance as a valuable tool in CAD/CAM systems. They are used to produce curves, which appear reasonably smooth at all scales. At present, many CAD/CAM systems employ B-spline curves as their major building blocks, since they can attain a number of mathematical properties. The main purpose of this paper is to present a class of newly constructed cubic trigonometric B-spline curves with a parameter. The proposed curves not only inherit all geometric properties of the standard cubic uniform B-spline curves, but also are 3 and can be adjusted by altering the value proper conditions, the -B-spline curves can also exactly represent ellipses and parabolas.  (e) Shape-adjustability: a family of curves of -B-spline basis functions controlled by parameter can be obtained. Figure 1 shows curves of the -B-spline basis functions with a = 0.3 (dotted lines), a = 0.4 (solid lines) and a = 0.5 (dashed lines).
Note that the value of the parameter a can actually be taken as any real number. But the -B-spline basis functions would not be nonnegative when 0.3 or 0.3 , which can be illustrated by drawing the curves of the basis functions. Thus, the parameter is taken as [0.3, 0.5] to ensure that the -B-spline basis functions are nonnegative.
Then, the following cubic trigonometric B-spline curves with a parameter ( -B-spline curves for short) -B-spline basis functions.

RESULTS AND DISCUSSION
From properties of the -B-spline basis functions, some properties of -B-spline curves can be obtained as follows: (a) Geometric invariance: because equation (2) is a vector function, shape of -B-spline curves is determined by the control points and parameter and has nothing to do with the coordinate system.  ... (7) By equation (4) ~ equation (7), then Equation (8) shows that the -B-spline curves are 3 .
(e) Shape-adjustablity ... (10) When a b , equation (10) is the expression of arcs of circle; when a b , equation (10) shows arcs of ellipse. Figure 4 shows the exact representation of arcs of ellipse by the -B-spline curves for ... (11) Equation (11) shows arcs of parabola. Figure 5 shows the exact representation of arcs of parabola by the -Bspline curves for 2 a and 1 b . curves can be a better approximation to the control polygon than the standard cubic uniform B-spline curves.

Remark 1. It is clear that
As mentioned above, shape of the -B-spline curves can be adjusted by the parameter even if the control points are kept unchanged. Hence, smoothness of the -B-spline curves is mainly determined by the parameter is chosen improperly, the corresponding -B-spline curves -B-spline curves with 0.4 for the same control points in Figure 3.
It is clear that the -B-spline curves in Figure 3 are smoother than those in Figure 6. Therefore, smoothing problem of the -B-spline curves needs to be considered when choosing value of the parameter .
According to smoothing criterion (Poliakoff, 1996),  To make the -B-spline curves as smooth as possible, one may need to determine the value of the parameter to minimise the energy value of the curves. Hence, an optimisation model for determining the value of the parameter can be obtained as,  Figure 7 shows the smoothest -B-spline curves for the same control points in Figure 3, where the value of the parameter found using equation (13) is 0.3375 .

CONCLUSION
The -B-spline curves presented in this paper not only have properties similar to the standard cubic uniform B-spline curves, but also are 3 and can be adjusted by altering the value of the parameter even if the control -B-spline curves can also exactly represent arcs of ellipse and parabola. In order to make sure the -B-spline curves are as smooth as possible, a method for determining the value of the parameter is presented. Because there are no differences in structure between the -B-spline curves and the standard cubic uniform B-spline curves, it is not -B-spline curves to the CAD/CAM systems that already use the corresponding standard cubic uniform B-spline curves.