New Cubic Trigonometric Bezier-Like Functions with Shape Parameter: Curvature and Its Spiral Segment

This work presents the new cubic trigonometric Bézier-type functions with shape parameter. Basis functions and the curve satisfy all properties of classical Bézier curve-like partition of unity, symmetric property, linear independent, geometric invariance, and convex hull property and have been proved. The C 3 and G 3 continuity conditions between two curve segments have also been achieved. To check the applicability of proposed functions, different types of open and closed curves have been constructed. The effect of shape parameter and control points has been observed. It is observed that, by decreasing the value of shape parameter, the curve moves toward the control polygon and vice versa. The CT-Bézier curve is closer to the cubic Bézier curve for a fixed value of shape parameter. The proposed CT-Bézier curve can be used to represent ellipse. Using proposed basis functions, we have constructed the spiral segment which is very useful to construct fair curves and desirable to design trajectories of mobile robots, highway, and railway routes’ designing.

Design of a Bézier-Profile Plate for Closely Spaced Multimodal Vibration Energy Harvesting

A cubic Bézier-profile plate for multimodal vibration energy harvesting was developed. The design of the plate was based on an optimization procedure in which the profile of the plate was optimized via the parameters of a cubic Bézier curve to meet the requirements. The multimodal energy harvesting of the plate exploited its first bending mode and its first twisting mode. The conversion of vibration energy into electrical energy was by electromagnetic induction with a magnet attached to a corner of the plate. These two closely spaced vibration modes achieved the multi-modal energy harvesting of the device. Prototypes of the device were manufactured using a numerical-control machining process. The experimental results were in good agreement with the design specifications. With the same base lengths, height, and thickness, the maximum von Mises stress of the proposed plate was much lower due to its bell-shaped profile. The cubic Bézier curve chosen for the plate profile was effective for design of the closely-spaced multimodal vibration energy harvester. With the flexibility of its controllable parametric curve, a high design freedom of the energy harvester with specified frequency ratios could be achieved.

On the Involute of the Cubic Bezier Curve by Using Matrix Representation in E3

In this study we have examined, involute of the cubic Bezier curve based on the control points with matrix form in E3. Frenet vector fields and also curvatures of involute of the cubic Bezier curve are examined based on the Frenet apparatus of the first cubic Bezier curve in E3.