A 3-point point quinary approximating subdivision schemes and its application in geometric modeling and computer graphics

. This article discusses the significance of a subdivision scheme with shape parameters in geometric modeling and computational geometry. A new recursive method for generating the mask of 3-point quinary approximating subdivision schemes (ASSs) is presented. The proposed subdivision scheme exhibits all the geometric properties and o ff ers better shape control than non-parametric subdivision schemes. The article includes several numerical examples to demonstrate the high practical value of the proposed schemes in geometric modeling and computer graphics.


Introduction
In the field of computational geometry and applied sciences, the use of Subdivision Surfaces (SSs) is limited due to their flaws.To reduce these deficiencies, a lot of work has been carried out by researchers [10][11][12][13][14][15][16].One of the shortcomings of non-parametric SSs is their representation in polynomial form.Therefore, many researchers are searching for solutions to this problem in non-polynomial function space.
Over the past few decades, researchers have explored the use of trigonometric functions for describing curves and surfaces.These SSs are now playing an important role in several areas including medicine, electronics, and animations.Hao et al. [17] worked on the conditions for shape control parameters of the SS to obtain convexity and produce C r (r = 7, 8, 9) limit curves.Zhijie [18] proposed a shape preserving technique for classical interpolating 4-point SS.Wang and Li [19] derived the convexity preserving conditions for the ternary 4-point interpolatory SS in [20].Albrecht and Romani, [21] discussed the convexity preserving conditions for the interpolatory SS with conic precision.Novara and Romani [22] presented the condition for assuring the convexity preservation of the 5-point interpolating SSs.Recently, computational geometry using SSs has received much attention.Mustafa et al. [23,24] constructed parametric schemes based on a new kind of Lagrange polynomial with a single shape parameter.Ghaffar et al. [25,26] formulated continuity conditions for the representation of free-form unified SSs with single shape control parameter.These freeform convoluted shapes can be attained by using shape control unified SSs.These newly defined approaches not only take over the advantages of non-parametric SSs but also resolve the issue of shape controlling of SSs with the help of tension control shape parameters.The proposed masks of the schemes provide different techniques to develop composite curves using higher smoothness conditions with straightforward and simple computation for proposed SSs because they are blended with linear polynomials rather than trigonometric functions.
In 2020, Ashraf et al. [27,28] published a new method for the generalization of SSs with a single shape parameter.Shahzad et al. [29] constructed a new numerical algorithm to calculate the subdivision depth of binary SSs with shape parameters.Hussain et al. [30] established an algorithm that generates 5-point approximating SS of varying arity with a single shape parameter.To address these shortcomings, researchers have worked on techniques such as involving shape adjusting parameters in the subdivision process and using non-polynomial function spaces.Trigonometric functions have been used for the description of curves and surfaces, and have found applications in several fields.Recent proposed research has focused on constructing new 3-point ASSs with single shape parameter with C 2 continuity.This scheme have potential applications in computer graphics, computer animation, and multimedia technology.Limit curves and their properties are also discussed.

Construction of 3-point quinary ASSs
The family of (3)−points quinary ASS can be obtained from the sets of the coefficients (called the masks of the SSs and represented by (α 5  3,t ) of following polynomial where From (1), we obtained different polynomials P 5 3,t (z) for each t.The superscript "5" of polynomials P 5  3,t (z) and α 5 3,t (z) stands for quinary schemes while lower subscript "3" means (3)-point schemes.The shape parameter in the Eq.(1) gives us the opportunity to select the order of continuity for proposed scheme up to C [ 5n− 3  2 ]−1 .For n = 2 in equation (1), we obtain, where t = 1, 2, 3, ......
From the coefficient of the polynomial (2), we can generate family of 3-point quinary approximating schemes one scheme for each t.
Substituting the values of u 3 , u 4 , u 5 and u 6 , we get The coefficients (mask) of the above polynomial are Putting n = 2 and t = 3 in Eq. ( 1), we get where u 2 = u 0 .Then, Substituting the values of u 1 and u 2 in above equation, we get The coefficients (mask) of the above polynomial are

Analysis of 3-point quinary scheme
Since the mask of the scheme ( 13) is given by The Laurent's Polynomial corresponding to this mask is given by Now by using the formula This implies Let S 1 be the SS corresponding to the Polynomial a 1 (z), then for C 0 continuity of scheme S , we require that ∥ 1 5 S 1 ∥ ∞ < 1.The norm infinity of the scheme S 1 is given by  .

This implies
This implies that S 1 is contractive, and S is C 0 continuity if and only if −1 5 < u 0 < 3 10 .By using This implies Let S 2 be the SS corresponding to the Polynomial a 1 (z), then for C 0 continuity of scheme S , we require that ∥ 1 5 S 2 ∥ ∞ < 1.The norm infinity of the scheme S 2 is given by This implies that S 2 is contractive, and S is C 1 continuity if and only if −2 25 < u 0 < 3 25 .Now by using (??), (??) and ( 15), we get This implies S 3 be the SS corresponding to the Polynomial a 1 (z), then for C 0 continuity of scheme S , we require that ∥ 1 5 S 3 ∥ ∞ < 1.The norm infinity of the scheme S 3 is given by

This implies
This implies that S 3 is Contractive, and S is C 2 continuous if and only if 0 < u 0 < 1 25 .No further continuity of the scheme exists.

Support of scheme
The mask of the 3-point quinary scheme has 15 terms.The middle value of our this mask is situated at the distance of 7 term from the left and right extreme non-zero coefficients.When we perform the first refinement level this distance reduces by 7  5 from each side.In this way, at every refinement step the sequence of the mask will continue to decrease by the same factor 7 5 × 1 5 .Consequently, after k subdivision levels, the distance of the extreme terms from the central term can be calculated by the formula, 7 So, when k → ∞, the total width of support of the limiting curve will be given by, 7 5 Graphical representation of support is showed in Figure 1.

Error bounds:
From the research conducted in Table 1 using [32] with χ = 0.1, we have calculated error bounds between the limit curve and the control polygon after k-fold subdivision of quinary approximating schemes.Our analysis of Figure .2 and Tables 1 has led us to the following conclusion: The error bounds of these schemes increase as the complexity (number of points involved to insert new points) of the SS increases.However, in the case of 3-point quinary approximating schemes, the error bounds remain constant and achieve their minimum value over 0 ≤ u 0 ≤ 1 10 .It is evident that the error bound of quinary approximating schemes gradually increases on both sides of the interval.

Geometrical significance and visual inspection
The presented schemes produce visually smooth curves that can be customized by altering the shape parameter within their domain.By adjusting this parameter, we can achieve remarkable beautification and attraction in the figures.For instance, Figures 3 showcase the graphs for 3point quinary approximating SSs with varying shape parameter values.We can easily obtain continuous curves that satisfy smoothness conditions and modify the subdivision curve as desired by adjusting the shape parameter, as exemplified in Figures 3.

Conclusion
In this section, we have analyzed the 3-point quinary approximating schemes and discussed their properties.We used the Laurent polynomial method to analyze the smoothness of the scheme.We also compared them with some existing schemes.Several examples are provided to illustrate that the proposed schemes give geometric designers a wide choice to generate smooth geometric models as per their needs.In terms of application in computer graphics, the resulting curves are smooth and highly suitable in designing any computer graphics shape and model.One possible future extension is the generalization of the proposed subdivision scheme in scattered data interpolation and image interpolation problems [33,34].