A cubic Bezier curve is a smooth parametric curve defined by four control points. The curve starts at the first point P0 and ends at the last point P3. The two middle points P1 and P2 act as handles that control the shape and direction of the curve without the curve necessarily passing through them.The mathematical formula for a cubic Bezier curve uses a parameter t that varies from 0 to 1. When t equals 0, the curve is at point P0. When t equals 1, the curve reaches P3. As t increases from 0 to 1, the point smoothly travels along the curve, with its position determined by a weighted combination of all four control points.Cubic Bezier curves have numerous applications in computer graphics and design. They are fundamental in font design, where each letter is composed of multiple Bezier curves. Vector graphics software like SVG and Adobe Illustrator rely heavily on Bezier curves to create scalable shapes. In animation, they define smooth motion paths. UI and UX designers use them to create elegant transitions and animations that feel natural and responsive.The real power of Bezier curves lies in their interactive control and flexibility. Designers can intuitively manipulate the curve by dragging control points. The curve remains smooth and continuous regardless of how the control points are positioned. This mathematical precision combined with visual intuition makes Bezier curves an essential tool. They are also computationally efficient, allowing real-time manipulation even in complex designs.In web development, cubic Bezier curves are fundamental for CSS animations and transitions. Developers use the cubic-bezier function to define custom timing functions that control how animations progress over time. Common easing functions like ease-in, ease-out, and ease-in-out are all based on Bezier curves. These curves map time to animation progress, creating natural-looking motion that enhances user experience. By adjusting the control points, developers can create unique animation effects that match their design vision.