G² cubic trigonometric spline for curve and surface fitting

Free-form curves and surfaces are frequently required in various industry for a wide range of applications in the fields of science and engineering. Nevertheless, there are difficulties in producing complex curves and surfaces with both smoothness and great precision. Thus, this thesis proposed a novel cubic trigonometric spline with a single shape parameter to address the given circumstances. The basis functions of cubic trigonometric spline are constructed based on Bezier-like functions, that satisfied all the properties needed. The presence of shape parameter plays an essential role in modifying the shape of the curve. Various shapes of curves and surfaces could be created even with the presence of limited shape parameter. Parametric and geometric continuity conditions have also been implemented and achieved to guarantee the smoothness of joining the piecewise curves and surfaces. There are C⁰, C¹, C², G⁰, G¹ and G² continuity. The presence of scalar factor in continuity conditions allows the degrees of freedom in handling the formation of curves and surfaces. The conditions for shape preservation of positive data set are derived. The process of modelling and handling complex product using continuity constraint is quick and easy to control, making it easier to meet the actual need even applying to real data. The mathematical outline of 2D boundary images of ‘ى, ‘epsilon’, and ‘delta’ are reconstructed using proposed cubic trigonometric spline with other three different cubic trigonometric spline in literatures. All the chosen cubic trigonometric spline shared same criteria regarding the type of spline, number of shape parameter, degrees of spline and having Bezier-like properties. The splines undergo the curve fitting process including corner points detection, control points calculation, curve fitting, and error calculations. The approximation errors are calculated by computing the distance from the actual image and the fitted curves to determine each spline's effectiveness. The performance of each spline is compared visually and numerically. The results show that the proposed functions provide higher accuracy representations of curves which are continuous, smooth and pleasant.