The construction of new basis functions for the Bézier or B-spline curve has been one of the most popular themes in recent studies in Computer Aided Geometric Design (CAGD). Implementing the new basis functions with shape parameters provides a different viewpoint on how new types of basis functions can develop complex curves and surfaces beyond restricted formulation. The wide selection of shape parameters allows more control over the shape of the curves and surfaces without altering their control points. However, interpolated parametric curves with higher degrees tend to overshoot in the process of curve fitting, making it difficult to control the optimal length of the curved trajectory. Thus, a new parameter needs to be created to overcome this constraint to produce free-form shapes of curves and surfaces while still preserving the basic properties of the Bézier curve. In this work, a general fractional Bézier curve with shape parameters and a fractional parameter is presented. Furthermore, parametric and geometric continuity between two generalized fractional Bézier curves is discussed in this paper, as well as demonstrating the effect of the fractional parameter of curves and surfaces. However, the conventional parametric and geometric continuity can only be applied to connect curves at the endpoints. Hence, a new type of continuity called fractional continuity is proposed to overcome this limitation. Thus, with the curve flexibility and adjustability provided by the generalized fractional Bézier curve, the construction of complex engineering curves and surfaces will be more efficient.

Secure image communication over public networks requires encryption methods that provide strong security guarantees while maintaining computational efficiency for practical deployment. Existing Elliptic Curve Cryptography Hill Cipher schemes offer efficient matrix-based encryption with compact key representation and low computational cost. However, their security performance is closely tied to elliptic curve parameter selection and relies on a single linear transformation, which may limit key diversity and diffusion capability. This paper presents a hybrid image encryption scheme that strengthens the conventional Elliptic Curve Cryptography Hill Cipher by incorporating a Bézier-like Coefficient Matrix as an additional transformation layer. The proposed matrix is generated through a controllable shape parameter, enabling the construction of diverse transformation matrices that expand the effective key space and act as diffuse to randomize the pixel values of ciphered image. The resulting two-stage encryption framework improves the diffusion process and enhances the complexity of the ciphered image while preserving computational efficiency. Security and performance are evaluated through key sensitivity analysis, information entropy, histogram distribution, differential attack resistance, statistical randomness, and robustness against noise interference. Experimental results show that the proposed method achieves near ideal entropy, uniform ciphered image distributions, high sensitivity to minor key variations, and stable image recovery under moderate noise conditions. Comparative analysis further demonstrates improved diffusion characteristics and enhanced resistance to statistical and differential attacks over the conventional Elliptic Curve Cryptography Hill Cipher. These results indicate that the proposed framework provides a secure and computationally efficient solution for image protection and secure multimedia communication.