Based on quintic trigonometric Bézier like basis functions, the biquintic Bézier surfaces are modeled with four shape parameters that not only possess the key properties of the traditional Bézier surface but also have exceptional shape adjustment. In order to construct Bézier like curves with shape parameters, we present a class of quintic trigonometric Bézier like basis functions, which is an extension of a traditional Bernstein basis. Then, according to these basis functions, we construct three different types of shape adjustable surfaces such as general surface, swept surface and swung surface. In addition to the application of the proposed method, we also discuss the shape adjustment of surfaces showing with curvature nephogram (with and without fixing the boundaries). However, the modeling examples shows that the suggested approach is efficient and easy to implement.

This paper introduces a new approach for the fabrication of generalized developable cubic trigonometric Bézier (GDCT-Bézier) surfaces with shape parameters to address the fundamental issue of local surface shape adjustment. The GDCT-Bézier surfaces are made by means of GDCT-Bézier-basis-function-based control planes and alter their shape by modifying the shape parameter value. The GDCT-Bézier surfaces are designed by maintaining the classic Bézier surface characteristics when the shape parameters take on different values. In addition, the terms are defined for creating a geodesic interpolating surface for the GDCT-Bézier surface. The conditions appropriate and suitable for $G^1$, Farin–Boehm $G^2$, and $G^2$ Beta continuity in two adjacent GDCT-Bézier surfaces are also created. Finally, a few important aspects of the newly formed surfaces and the influence of the shape parameters are discussed. The modeling example shows that the proposed approach succeeds and can also significantly improve the capability of solving problems in design engineering.

B-Spline is an approximating curve, its shape is determined by the control points but it also has parametric nature. To interpolate data points, various works have been done by the previous researcher in B-Spline parameterization. The proposed study is the extension in the B-spline interpolation technique using arbitrary end derivatives. In this paper, we present a new way for interpolating cubic B-Spline curve by taking first and second derivative at endpoints and only first derivative at inner points. An algorithm is presented for interpolating data points. The algorithm computes knot values for parameterization methods. These knot values are used in constructing a matrix of B-Spline basis function and derivative of the basis function. Then, we solved it for unknown control points by using the LU decomposition method, such that the curve will pass through the original data points. For interpolation, the parameterization method used in this paper is exponential parameterization. The main advantage of this concept of derivatives is that we can generate different shapes of curves by setting different direction at all points. As an application, our method is applied in a curve reconstruction on road map with data points and driving directions.