A cubic Bezier curve is a parametric curve defined by four control points. The curve starts at the first point P0 and ends at the last point P3, while the middle two points P1 and P2 control the shape of the curve. This type of curve is widely used in computer graphics, animation, and font design because it provides smooth and flexible shapes.The cubic Bezier curve is defined by a parametric equation where t ranges from 0 to 1. The formula is B of t equals one minus t cubed times P0, plus three times one minus t squared times t times P1, plus three times one minus t times t squared times P2, plus t cubed times P3. When t equals 0, the curve is at point P0. When t equals 1, the curve is at point P3. As t varies from 0 to 1, the point traces out the entire curve.The cubic Bezier curve can be constructed through a recursive linear interpolation process known as De Casteljau's algorithm. First, we interpolate between adjacent control points to get three intermediate points Q0, Q1, and Q2. Then, we interpolate between these points to get two more points R0 and R1. Finally, interpolating between R0 and R1 gives us the point B of t on the curve. As parameter t changes, this construction process traces out the entire Bezier curve.Cubic Bezier curves have several important properties. First, the curve always starts at P0 and ends at P3, which are called anchor points. Second, the curve is tangent to the line P0P1 at the starting point P0, and tangent to the line P2P3 at the ending point P3. This means the control points P1 and P2 determine the initial and final directions of the curve. Third, the curve always lies within the convex hull formed by the four control points, ensuring predictable behavior. Finally, moving any control point changes the curve shape smoothly, making it easy to design and adjust curves interactively.