On the Bicubic Bézier Surfaces and Paraboloids in E³

Bicubic Bézier surfaces are fundamental in the world of computer graphics, computer-aided design (CAD), and animation because they offer a powerful balance of flexibility, smoothness, and control. In addition to being compatible with other surface representations, they have attracted the attention of scientists due to their mathematical strength and understandability, and many studies have been conducted on this subject. In this paper, first we examine the matrix representation of bicubic Bézier surfaces whose control points lie in E³. Second, as examples, we consider elliptic and hyperbolic paraboloids as bicubic Bézier surfaces. Finally, we present a method for determining the control points of a given elliptic paraboloid, hyperbolic paraboloid, and parabolic cylinder as bicubic Bézier surfaces.