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A geometric modeling method based on TH-type uniform B-splines which are composed of trigonometric and hyperbolic polynomial with parameters is introduced in this paper. The new splines possess many important properties of quadratic and cubic B-splines. Taking different values of the parameters, one can not only locally adjust the shape of the curves, but also change the type of some segments of a curve between trigonometric and hyperbolic functions as well. The given curves can also interpolate directly control polygon locally by selecting special parameters. Moreover, the introduced splines can represent some quadratic curves and transcendental curves with selecting proper control points and parameters.

B-splines are used as an important geometric modeling tool in computer aided geometric design (CAGD). However, there are still several limitations on B-splines in practical applications [

By comparing T-type uniform B-splines and H-type uniform B-splines, we found that T-type uniform B-splines are located on one side of the B-spline, and H-type uniform B-splines are located on the other side of the B-spline. Therefore, one thinks if the two different curves can be unified to produce new blending splines, then the new curve will have more plentiful modeling power. In order to construct more flexible curves for curves and surface modeling, Zhang et al. [

In this paper, we present a new geometric modeling method based on two kinds of TH-type uniform B-splines which are composed of hyperbolic and trigonometric functions. The introduced spline has the following features: (1) the new spline curves can be adjusted totally or locally. (2) The given curves can switch into T-type B-spline curves or H-type B-spline curves when the parameter is equal to 0 or 1. (3) Without solving the system of equations, the new curves can interpolate certain control points directly. (4) The TH-type B-spline curves can be used to represent some conics and transcendental curves with the parameters and control points chosen properly.

The rest of this paper is organized as follows. In Sections

Given

The above functions have the following properties.

Partition of unity:

Symmetry:

Nonnegativity: if

(i) and (ii) are easy to be proved by simple computation. Next we will prove (iii).

By direct computation, we have

From (ii), we have

Let

Given control points

By a similar method, we may define the bases of cubic TH-type B-spline.

For

It is easy to prove that the basis functions of cubic TH-type B-spline have the same properties: nonnegativity, partition of unity, and symmetry.

Given control points

According to the properties of the basis functions and definition, it is easy to get the following properties of curves (

For the uniform knots, the curves (

For the curve (

For the curves (

This implies the theorem.

Local adjustable quadratic uniform TH-type spline curves.

Local adjustable cubic uniform TH-type spline curves.

Obviously, when all parameters

Local interpolating quadratic TH-type spline curves.

Local interpolating cubic uniform TH-type spline curves.

When the parameters

The ellipse and hyperbola are the most common in the conic curve. If the control points and the parameters are selected properly, the curves (

Given the uniform knots, for the quadratic T-type B-spline curve, we take the coordinates of the points

The representation of ellipse with quadratic (a) and cubic (b) T-type B-spline curves.

For the cubic H-type uniform B-spline curves, we take

The representation of hyperbola with quadratic (a) and cubic (b) H-type B-spline curves.

In this section, we can represent the transcendental curves with the uniform TH-type B-splines, such as cycloid and catenary.

When parameters

The representation of cycloid with cubic T-type B-spline curves.

Similarly, when taking

The representation of catenary with cubic H-type B-spline curves.

From the last section, we see, letting the parameter be equal to 0 or 1, the types of the curves can be switched easily. So, by selecting control points and parameters properly, we can represent different type curve segments among a blending curve. In Figure

A closed

Figure

An open

The author declares that there is no conflict of interests regarding the publication of this paper.

This work was funded by the Natural Science Foundation of Anhui Province of China under Grant no. 1208085MA15, the Key Project Foundation of Scientific Research, Education Department of Anhui Province under Grant no. KJ2014ZD30, and Key Construction Disciplines Foundation of Hefei University under Grant no. 2014XK08.