Construction and Deformation of Curves and Surfaces Based on α–Sh Basis Functions with Shape Parameters

Curve and surface modeling techniques have long been essential in computer graphics research. However, many existing methods for curve and surface fitting, as well as deformation, have limitations, such as challenges in representing certain special curve forms and a lack of control over the deformation process. Therefore, there is a need for a method that accurately represents specific curves and surfaces while allowing for a more intuitive and straightforward deformation implementation. To address this, the paper proposes a new method for constructing curves and surfaces and for their deformation. First, a shape-controlled basis function, termed the α-sh basis function, is defined in the basis vector space 1,t,t²,...,tⁿ⁻², sinh t, cosh t[bracerightbig]. Next, the favorable properties of the α-sh basis function are analyzed and proven, demonstrating its feasibility for curve and surface fitting. Using this basis function, α-sh Bézier curves and αβ-sh Bézier surfaces are defined, and their properties are thoroughly analyzed and proven. Finally, by adjusting the shape control parameters, the deformation of curves and surfaces can be achieved. The proposed method also enables the representation of special curves, such as circles, and allows for their deformation. The paper concludes with examples of curves and surfaces, visualizations of their deformation effects, and potential applications in practical industrial design.