Shape-Preserving Positive Trigonometric Spline Curves

This paper deliberates on the development of new smooth shape-preserving schemes. These schemes are also demonstrated with data in the form of shape-preserving trigonometric spline curves. For this persistence, a trigonometric spline function is developed, which also nourishes all the fundamental geometric properties of Bézier function as well. The developed trigonometric spline function comprises two parameters αi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha_{i} $$\end{document} and βi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \beta_{i} $$\end{document}, which ensures flexible tangents at the end points of each subinterval. Furthermore, constraints are derived on βi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \beta_{i} $$\end{document} to generate the shape-preserving trigonometric spline curves, whereas αi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha_{i} $$\end{document} is any positive real number used for the modification of shape-preserving trigonometric spline curves. The error approximation of the developed function is Ohi3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ O\left( {h_{i}^{3} } \right) $$\end{document}.