A Rational Quadratic Trigonometric Spline (RQTS) as a Superior Surrogate to Rational Cubic Spline (RCS) with the Purpose of Designing

Round 1

Reviewer 1 Report

 

Review Report

Manuscript No.: applsci-1633653

 

Dear Respected Editor,

I have read this paper in detail and the following are my opinion and observation about the paper:

1-Authors should improve the English as well as the presentation of this paper. There are many typos in the paper and the punctuations after each equation must be revised.

2- Authors should explain more about the novelty of this work in the introduction, indeed, several studies have addressed the enhanced modified simple equation method. 
3- The authors must clearly indicate what is the difference between the preset work history and other research work.
4-The goals, the general motivations, and the highlights of the problem and results, as well as the use of this approach are not well explained.

5- The abstract must not include any abbreviations and the keywords should not include any words in the title.

6- The authors should explain how the initial guess was chosen and what kind of stopping criteria was considered for nonlinear solver used for solving the test problems.

7- The CPU time for the execution of the programs for solving numerical examples should be presented in the numerical illustration section of the paper in order to show the computational efficiency of present method. The authors should compare the CPU time of the proposed method with other methods.

8- The authors need to present a detailed comparison with the following important work: Physica Scripta 96 (10), 104001, 2021.

9- The computational order of convergence of the proposed method is not provided in the paper. The authors should validate numerical result with the theoretical estimate.

10-Conclusion part of the paper must be extended a little more.

After these modifications, this paper may be accepted.
With my best regards

Author Response

Dear Reviewer,

Kindly find the attached file of author's rebuttal.

Author Response File: Author Response.pdf

Reviewer 2 Report

The paper deals with rational quadratic trigonometric splines. I have several strong objections against the proposed manuscript.   First of all, it is claimed that trigonometric splines are superior to the classical polynomial and rational counterparts. I strongly disagree with this claim. There can be some advantages in using trigonometric functions, but there are many disadvantages that are not mentioned by the authors, either intentionally or by lack of knowledge.   In particular, function evaluation and differentiation is a lot simpler for polynomial splines. One can use de Casteljau algorithm on a particular polynomial/Bezier segment to quickly evaluate the function values and their derivatives. This is a lot more difficult for trigonometric functions as one has to use infinite series to approximate sin and cos function, and these series have to be truncated for a sufficiently large N to guarantee the desired precision (e.g. machine precision). Such a evaluation and differentiation is therefore a lot slower than evaluation via de Casteljau algorithm.   The paper speaks about "best geometric properties", but properties such as partition of unity, convex hull, affine invariance, or variation diminishing properties of course hold also for Bezier/splines so there is nothing new that would outperform the traditional modeling tools.   Moreover, some of these properties are not proven correctly. E.g., the affine invariance in the classical sense of Bezier curves says that applying an affine mapping on the control points, and constructing a curve, gives the same result as constructing a curve and applying the affine map afterwards. This is not what is done in this paper. The proof of Proposition 1 applies an affine mapping on the coefficients of the basis functions, and not on the control points, which are two different things. Btw. the control points are defined synthetically and have no good geometrical meaning, unlike Bezier/spline/NURBS control points.   Btw., what is shown in fig.2 is not affine invariance as all the mapping are homotheties (very special affine mapping), and not general affine mappings.   I am also not convinced that the proof of Theorem 2 is correct. It is rather long, difficult to read, and with some unclear estimates. This should be rewritten, but also supported by some numerical experiments where one could clearly see the O(h^3) behavior of the approximation error.   There is no clear advantage over classical Bezier/splines/NURBS, e.g., in Fig.7. All these shapes can be easily constructed using the traditional tools. And again, for a lot smaller computational cost (see my comments in at the very beginning).   Last but not least, the state-of-the-art should be completely rewritten. Out of 19, there are 16 self-references of the authors that is completely unacceptable. It makes the impression that the authors form exclusively the state of the art, which is by far not true.   There are plenty of papers on curve modeling, trigonometric, polynomial, and/or rational, including shape parameters, see e.g.,   Lyche, T. and Winther, R., 1979. A stable recurrence relation for trigonometric B-splines. Journal of Approximation theory, 25(3), pp.266-279.   Pelosi, F., Farouki, R.T., Manni, C. and Sestini, A., 2005. Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics. Advances in Computational Mathematics, 22(4), pp.325-352.   Ait-Haddou, R., and Barton, M. Constrained multi-degree reduction with respect to Jacobi norms. Computer Aided Geometric Design 42 (2016): 23-30.   Koch, P.E., Lyche, T., Neamtu, M. and Schumaker, L.L., 1995. Control curves and knot insertion for trigonometric splines. Advances in Computational Mathematics, 3(4), pp.405-424.   Lyche, T., Schumaker, L.L. and Stanley, S., 1998. Quasi-interpolants based on trigonometric splines. Journal of Approximation theory, 95(2), pp.280-309.   González, C., Albrecht, G., Paluszny, M. and Lentini, M., 2018. Design of $$ C^ 2$$ algebraic-trigonometric pythagorean hodograph splines with shape parameters. Computational and Applied Mathematics, 37(2), pp.1472-1495.   BiBi, S., Abbas, M., Miura, K.T. and Misro, M.Y., 2020. Geometric modeling of novel generalized hybrid trigonometric Bézier-like curve with shape parameters and its applications. Mathematics, 8(6), p.967.   Usman, M., Abbas, M. and Miura, K.T., 2020. Some engineering applications of new trigonometric cubic Bézier-like curves to free-form complex curve modeling. Journal of Advanced Mechanical Design, Systems, and Manufacturing, 14(4), pp.JAMDSM0048-JAMDSM0048.   and the top textbook   Farin, G., Hoschek, J. and Kim, M.S. eds., 2002. Handbook of computer aided geometric design. Elsevier.   and many more relevant references cited therein. I strongly recommend the authors to significantly reduce the number of their own papers and to cite only those that are really relevant.   Moreover, there are some papers among the references that seem to have absolutely no relation with the topic of this paper, see e.g. [19]. Such a reference should be removed and only relevant works to the topic should be cited.

Author Response

Dear Reviewer,

Kindly find the attached file of author's rebuttal.

Author Response File: Author Response.pdf

Reviewer 3 Report

The problem is timely and interesting. I recommend the publication of the manuscript after the following revisions are properly made:

 

1) The aim of the paper is not completely well-specified. The authors could specify more this aspect in the abstract and in the introduction of the manuscript.

2) Please, check if all the symbols used in the various equations are well-defined in the text.

3) English should be enhanced throughout the manuscript to eliminate grammatical errors and misprints.

4) The advantage of the used method should be more discussed.

5) For general readers, authors are encouraged to discuss other kinds of methods such as: [(a) "Dynamic stability/instability simulation of the rotary size-dependent functionally graded microsystem”; (b) “Non-polynomial framework for stress and strain response of the FG-GPLRC disk using three-dimensional refined higher-order theory"; (c) “Frequency simulation of viscoelastic multi-phase reinforced fully symmetric systems”; (d) “On the vibrations of the imperfect sandwich higher-order disk with a lactic core using generalize differential quadrature method”; (e) “Chaotic simulation of the multi-phase reinforced thermo-elastic disk using GDQM”].

 

6) In conclusion, give only main findings of your research with an appropriate value.

Author Response

Dear Reviewer,

Kindly find the attached file of author's rebuttal.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

No other comments.

 

I suggest the acceptance of the paper.