Generalized Riemann-Liouville Fractional Bézier Curve and Its Applications in Engineering Surface

Modeling of free-form curves and surfaces is vital in the manufacturing industry and engineering. Curves and surfaces that have flexible shapes and adjustable lengths and sizes are a necessity to fulfill the needs of the manufacturing industry. Hence, researchers develop numerous aesthetic Bézier curves and surfaces to have such flexibility and adjustability. In this paper, the generalized Riemann–Liouville fractional Bézier curves and surfaces are proposed in the modeling of complex surfaces. The generalized Riemann–Liouville fractional Bézier curves and surfaces have two outstanding parameters: shape and fractional parameters. Shape parameters are utilized to change the shape of the curves and surfaces without altering the control points hence, adding flexibility in controlling the shape. While fractional parameters are utilized in curves and surfaces’ length and size adjustability. By adjusting the sizes of surfaces via fractional parameters, surfaces with optimal sizes can be modeled. In this paper, five types of engineering surfaces will be modeled, namely ruled surface, swept surface, swung surface, rotation surface, and coons patch. The geometric effects of the implementation of shape and fractional parameters to these engineering surfaces will also be analyzed, thus proving the generalized Riemann–Liouville fractional Bézier surfaces is an excellent tool in designing complex surfaces.