G^1 Cubic Trigonometric Triangular Spline with Rational Basis Functions

Background: Digitization of an object from visual to data point will help researchers manipulate the data for further use – particularly in reconstructing a surface. The use of trigonometric function to generate a triangular surface are able to offer smooth surface with flexibility because it has the advantages of derivation and cyclability. There are studies that provides trigonometric basis functions for curves and surfaces with certain range of shape parameters. Methods: This study uses cubic trigonometric triangular spline with rational basis functions to construct G^1 continuity for curves and surfaces. Results: The basis is proven suitable for curve and surface design because it possesses blossoms which exist if and only if the space 〖DT〗_(λ,μ) is an EC-space on [0,π⁄2]. The G^1 continuity design is constructed and the results are illustrated in this study. Conclusion: The study develops four univariate basis functions for curves and extends them to ten basis functions for surfaces using a triangular domain. These basis functions exhibit desirable traits such as end-point property, non-negativity, partition of unity, monotonicity, linear independence, and symmetry. While this construction marks a foundational step, further research is needed to meet broader geometric design industry needs.