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A class of algebraic-trigonometric blended splines. (English) Zbl 1204.65167

Summary: This paper presents a new kind of algebraic-trigonometric blended spline curve, called \(xyB\) curves, generated over the space \(\{1,t,\sin t,\cos t,\sin^{2}t,\sin^{3}t,\cos^{3}t\}\). The new curves not only inherit most properties of usual cubic B-spline curves in polynomial space, but also enjoy some other advantageous properties for modeling. For given control points, the shape of the new curves can be adjusted by using the parameters \(x\) and \(y\). When the control points and the parameters are chosen appropriately, the new curves can represent some conics and transcendental curves. In addition, we present methods of constructing an interpolation \(xy\)B-spline curve and an \(xy\)B-spline curve which is tangent to the given control polygon. The generation of tensor product surfaces by these new spline curves is straightforward. Many properties of the curves can be easily extended to the surfaces. The new surfaces can exactly represent the rotation surfaces as well as the surfaces with elliptical or circular sections.

MSC:

65T40 Numerical methods for trigonometric approximation and interpolation
65D07 Numerical computation using splines
41A15 Spline approximation
65D05 Numerical interpolation
42A10 Trigonometric approximation
42A15 Trigonometric interpolation
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