# Geometric Modelling of a Family of 4-Point Ternary Approximating Subdivision Scheme U_φ with Visual Performance

## DOI:

https://doi.org/10.21015/vtm.v12i1.1787## Keywords:

Computer-aided Geometric Design, Ternary Subdivision Scheme, Geometric Modelling, approximating Algorithms, Control Polygon, Limit Curve.## Abstract

Making signals better than noise in communication has always been challenging for scientists. Researchers have been working on it in different ways. The computer-aided geometric design is a new research field emerging from the collaboration of computer algorithms and mathematical logic towards curve designing, in which the subdivision schemes used have a key position due to their flexible and smooth behaviour. Using parameters in these schemes allows for increased control over designing. A parameterized framework for generating a wide range of subdivision surfaces with tunable degrees of shape control is presented in the family of schemes. The properties of the proposed family make it suitable for use in isogeometric analysis, computer animation, and geometric modelling. The purpose of this paper is to construct and analyze a family of 4-point ternary subdivision schemes to smooth the curves based on the Laurant polynomial. This family is generated by tuning the weight parameter. The scheme is analysed for its different properties. The scheme has continuity. Visual performance of the subdivision scheme is also provided as an application of this proposed study.

## References

Hussain, S.M., Rehman, A.U., Baleanu, D., Nisar, K.S., Ghaffar, A. and Abdul Karim, S.A., 2020. Generalized 5-point approximating subdivision scheme of varying arity. *Mathematics*, 8(4), p.474.

Reif, U. and Sabin, M.A., 2019. Old problems and new challenges in subdivision. *Journal of Computational and Applied Mathematics*, 349, pp.523-531.

Liu, Y., Shou, H. and Ji, K., 2022. Review of subdivision schemes and their applications. *Recent Patents on Engineering*, 16(4), pp.50-62.

Asghar, M. and Mustafa, G., 2019. A family of binary approximating subdivision schemes based on binomial distribution. *Mehran University Research Journal of Engineering & Technology*, 38(4), pp.1087-1100.

Asghar, M., Iqbal, M.J. and Mustafa, G., 2019. A family of high continuity subdivision schemes based on probability distribution. *Mehran University Research Journal of Engineering & Technology*, 38(2), pp.389-398.

Zhang, L., Ma, H., Tang, S. and Tan, J., 2019. A combined approximating and interpolating ternary 4-point subdivision scheme. *Journal of Computational and Applied Mathematics*, 349, pp.563-578.

Abdul Karim, S.A., Khan, F., Mustafa, G., Shahzad, A. and Asghar, M., 2023. An efficient computational approach for computing subdivision depth of non-stationary binary subdivision schemes. *Mathematics*, 11(11), p.2449.

Yang, H., Kim, K. and Yoon, J., 2024. A family of $C^2$ four-point stationary subdivision schemes with fourth-order accuracy and shape-preserving properties. *Journal of Computational and Applied Mathematics*, p.115843.

Ashraf, P., Nawaz, B., Baleanu, D., Nisar, K.S., Ghaffar, A., Ahmed Khan, M.A. and Akram, S., 2020. Analysis of geometric properties of ternary four-point rational interpolating subdivision scheme. *Mathematics*, 8(3), p.338.

Certainly! Here are the remaining references formatted in Harvard style with counting 10-22 in square brackets:

Shahzad, A., Khan, F., Ghaffar, A., Mustafa, G., Nisar, K.S. and Baleanu, D., 2020. A novel numerical algorithm to estimate the subdivision depth of binary subdivision schemes. *Symmetry*, 12(1), p.66.

Ko, K.P., Lee, B.G. and Yoon, G.J., 2007. A ternary 4-point approximating subdivision scheme. *Applied Mathematics and Computation*, 190(2), pp.1563-1573.

Ko, K.P., Lee, B.G. and Yoon, G.J., 2007. A ternary 4-point approximating subdivision scheme. *Applied Mathematics and Computation*, 190(2), pp.1563-1573.

Mustafa, G., Ghaffar, A., Khan, F., et al., 2011. The odd-point ternary approximating schemes. *American Journal of Computational Mathematics*, 1(02), p.111.

Mustafa, G., Asghar, M., Ali, S., Afzal, A. and Liu, J.B., 2021. A family of integer-point ternary parametric subdivision schemes. *Journal of Mathematics*, 2021, pp.1-10.

Tariq, H.M., Hameed, R. and Mustafa, G., 2021. A new paradigm to design a class of combined ternary subdivision schemes. *Journal of Mathematics*, 2021, pp.1-19.

Mustafa, R., Iqbal, M.T., Mustafa, G. and Abdul Karim, S.A., 2024. Advancing curve and surface images modeling with two-parameters polynomial-based quaternary subdivision schemes. In: *ITM Web of Conferences*, volume 63, page 01013. EDP Sciences.

Khan, F., Mustafa, G., Shahzad, A., Baleanu, D. and Al-Qurashi, M.M., 2020. A computational method for subdivision depth of ternary schemes. *Mathematics*, 8(5), p.817.

Akram, G., Khan, M.A.U., Gobithaasan, R.U., Sadaf, M., Abbas, M., et al., 2023. Convexity and monotonicity preservation of ternary 4-point approximating subdivision scheme. *Journal of Mathematics*, 2023, pp.1-10.

Asghar, M., Mustafa, G., et al., 2018. Family of $α$-ary univariate subdivision schemes generated by Laurent polynomial. *Mathematical Problems in Engineering*, 2018, pp.1-10.

Hassan, M.F., Ivrissimitzis, I.P., Dodgson, N.A. and Sabin, M.A., 2002. An interpolating 4-point $C^2$ ternary stationary subdivision scheme. *Computer Aided Geometric Design*, 19(1), pp.1-18.

Rehan, K. and Siddiqi, S.S., 2015. A family of ternary subdivision schemes for curves. *Applied Mathematics and Computation*, 270, pp.114-123.

Ghaffar, A., Mustafa, G., Qin, K., et al., 2013. The 4-point $alpha$-ary approximating subdivision scheme. *Open Journal of Applied Sciences*, 3(1), pp.106-111.

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*VFAST Transactions on Mathematics*,

*12*(1), 290–310. https://doi.org/10.21015/vtm.v12i1.1787

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