Objective

2

The construction of B-spline basis functions has always been the focus of computer-aided design. The purpose of its research is mainly to solve the problem that the curve generated by the traditional method is fixed relative to the control vertices. The transformation form mainly incorporates the shape into the constructed basis function parameters to increase the flexibility of the curve

that is

to introduce free parameters to the expression of the classic Bernstein basis function or the extended Bernstein basis function

and adjusts the value of the parameter to adjust the shape of the curve. In recent years

researchers have proposed a large number of B-spline improvements

and they are mainly focused on two function spaces

namely

polynomial function space and trigonometric function space. The spline basis functions constructed in these two function spaces have their own advantages in addition to local adjustments to the corresponding curves. The spline curve constructed in the polynomial function space can be degenerated into a classic B-spline curve and has the advantages of simple calculation. Conversely

the basis function constructed in the trigonometric function space has the advantage of the derivation and cyclability of the trigonometric function. Both have high-order continuity

enabling the accurate representation of circle

ellipse

parabola

sine

cosine

cylindrical spiral

etc. The main purpose of this study is to combine the advantages of constructing spline basis functions in these two function spaces and use the weighting method to integrate the basis functions constructed in the two function spaces. The newly introduced weighting factors can be used as global parameters to make new extensions

and the flexibility of the curve is further enhanced. However

from the above two perspectives

some defects still exist. The reason is that the curve is edited and processed through the control vertices of the model

similar to the traditional method. When the control vertices are given

the curve is relatively fixed.

Method

2

First

construct a set of cubic rational basis functions in the polynomial function space and prove that they are fully positive. Then

use the weighting idea to weigh the newly constructed cubic rational basis functions and the cubic triangular basis functions constructed and new Bernstein basis and prove that it retains all the good properties of the classic Bernstein basis functions. Subsequently

the new extended basis is linearly combined to obtain the non-uniform cubic weighted B-spline basis

and its properties are studied. Finally

the definition and properties of the corresponding cubic spline curve are given

and the application of the new curve is given based on it.

Result

2

Experiments show that the curve constructed by the weighting method in this study retains the respective advantages of the polynomial function space and the trigonometric function space and also has local adjustment. At the same time

the introduced weighting factor also strengthens its global adjustment and further enhances the flexibility of the generated curve. It can improve the shortcomings of changing the curve shape only by adjusting the control vertices.

Conclusion

2

In this study

the new

λαβ

-B-spline curve constructed by the weighting method has a structure similar to the classic B-spline curve while retaining the same properties as the classic B-spline curve

such as convex hull

symmetry

geometric invariance

and change. In addition

the curve constructed by the weighting method in this paper has some unique advantages: first

it has two local shape parameters that can be adjusted locally; second

the weighting factor introduced can be used as a global parameter

and the generated

λαβ

-B-spline curve can perform global adjustment; third

the

λαβ

-B-spline curve constructed by the weighting method retains the respective advantages of the polynomial function space and the trigonometric function space. The results show that the shape parameter selection scheme of the constructed curve is correct and effective

which reflects the superiority of the method in this study over other similar methods in the literature. In addition

the construction method with parameter expansion base given in this study and the selection method of shape parameters are general

which can be extended to construct B-spline and triangle surfaces with shape parameters.