What are the uses of Bezier Curves and Bezier Surfaces? Explain the Mathematical expression of Bezier Curves. Write the properties of the Bezier curves, prove all properties. Discuss the Parametric Continuities and Geometric Continuities of Bezier Curves, with suitable expressions, equations and examples. Explain the purpose of control points in Bezier, a Cubic Bezier curve has control points P0 (0, 0); P1 (5, 40); P2 (40, 5); P3 (50, 15). Determine 2 more points on the same Bezier curve. Draw a rough sketch of the curve and show coordinates of various points on it?

Bezier curves and surfaces are fundamental tools in computer graphics and multimedia for creating smooth and aesthetically pleasing shapes. They offer versatility and control over the shape of curves and surfaces, making them widely used in various applications like animation, CAD (Computer-Aided Design), and font design.

Uses of Bezier Curves and Bezier Surfaces:
- Bezier Curves: They are used to represent paths and shapes in 2D graphics, such as drawing smooth curves for fonts, shapes in vector graphics, and animation paths.
- Bezier Surfaces: These are extensions of Bezier curves into higher dimensions and are used to create smooth surfaces in 3D modeling, such as in CAD _________ ______ ______ ______ ______ __________ _____ _______ _________ ________.
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