Computer Aided Geometric Design (CAGD) deals with studies of curves and surfaces used in computer graphics, data structure, and computational algebra. In CAGD, geometric shapes are related to the mathematical representations that satisfy approximation and interpolation properties of curves and surfaces. Surface modeling is one of the important studies in the fields of CAGD and computer graphics. It links mathematical sciences with computer science and engineering such as the animation industry, automotive and industrial design, aerospace, mechanical engineering, and numerical computing. Subdivision is an interesting subject and one of the common tools in CAGD, which provides an elegant way for the description of curves and surfaces modeling. Initially, Rham [
1] worked on subdivision schemes and made a scheme which generates a function with the first derivative. Similarly, Chaikin started work and used subdivision scheme to design a curve [
2]. Subdivision schemes gained importance when scientists generalized the tensor product in an arbitrary topology. Doo and Catmull used the subdivision schemes to establish surface design and control meshes in an arbitrary topology [
3,
4]. Deslauriers and Dubuc formed a 4-point scheme [
5]. Later, Dyn et al. [
6] generalized the scheme of Dubuc and Deslauriers, known as the butterfly scheme, which is based on approximated schemes. Cai used a 4-point scheme with non-uniform control points to calculate convergence and error estimation. He illustrated that the curves and surfaces generated from 4-point schemes gave better results [
7]. Hassan et al. [
8,
9] worked on arity and number of control points, whereas Mustafa and Xuefeng [
10] worked on the scheme of Bajaj with new a parameter which controls the shape of models and gave more flexibility to design a model over the soft and rough mesh network. Similarly, Siddiqui and Ahmad [
11] presented a 6-point subdivision scheme that gives better smoothness. Moreover, Hormann and Sabin [
12] produced a family of subdivision schemes to calculate support size, Hölder regularity, precision set, and degree of polynomial curve. Khan and Mustafa [
13] calculated an interpolating 6-point subdivision scheme for complex eigenvalues as well as worked on an approximating 4-point subdivision scheme. They showed that their scheme has higher smoothness and small support size as compared to other 4-point schemes [
14]. Mustafa et al. [
15] worked on the
m-point approximating subdivision schemes and illustrated that their schemes have higher smoothness as compared to other subdivision schemes. Siddiqi and Rehan [
16] worked on a 4-point binary scheme to generate the family of curves. They introduced a scheme for
continuity to generate a curve called corner cutting. Mustafa et al. [
17] further worked on odd-point ternary approximating subdivision schemes and developed a formula to generalize them. Later, Ghaffar et al. [
18] considered 3-point approximating subdivision schemes and observed that the given approach is more universal and is applied to schemes of arbitrary arity. Ghaffar et al. [
19] introduced a general formula for 4-point
a-ary approximating subdivision scheme for curve designing for any arity
.
In addition, Mustafa et al. [
20] worked over odd point ternary families of approximating subdivision schemes and showed that their schemes have high smoothness. They also worked on subdivision regularization, in which they showed that unified frame work can work well for both curve fitting and noise removal. They generalized unified families of interpolating subdivision schemes of
-point and
-point
p-ary which generate Lagrange’s polynomial for
and
, presented in [
21]. In 2013, Younus and Siddiqi [
22] established an algorithm based on Quaternary-point for
approximating subdivision scheme which has high smoothness and small support. Rehan et al. [
23] discussed the continuity of a new class of 3-point ternary schemes and generated limiting curves using the proposed schemes. They also proposed a 4-point ternary scheme which creates
interpolating and
approximating limiting curves, described in [
24]. For other recent work on this topic, we may refer to [
25,
26,
27,
28,
29] and references therein.
The above-mentioned literature shows limited knowledge about the arity of the SSs. This motivated us to construct a unified 5-point approximating SS of varying arity with the shape controlling parameter. To show the performance of the schemes, we analyze the geometric properties such as continuity, Hölder regularity, and Limit stencils. Moreover, the limit curves with the specific value of shape control parameter
w are depicted by the significant application of derived conditions on the initial data. The rest of the paper is organized as follows. The preliminaries regarding SSs are presented in
Section 2. In
Section 3, we analyze the geometric properties of the proposed schemes. The results and discussion are presented in
Section 4. Some example are considered in this section to show the efficiency of the schemes. Finally, the concluding remarks are given in final section.