A Bézier curve is a parametric curve widely used in computer graphics and design. It is defined by a set of control points that determine the shape of the curve. The curve passes through the first and last control points, called anchor points, while the intermediate points pull and guide the curve's shape without the curve necessarily passing through them.
There are different types of Bézier curves based on the number of control points. A linear Bézier curve uses two control points and creates a straight line. A quadratic Bézier curve uses three control points and can create smooth curved paths. The parameter t ranges from zero to one, determining how far along the curve we are.
Cubic Bézier curves are the most commonly used type in computer graphics and design software. They use four control points and provide excellent control over the curve's shape. The first and last points are anchor points that the curve passes through, while the middle two points act as control handles that pull the curve in their direction. This gives designers precise control over both the curve's path and its tangent directions.
The beauty of Bézier curves lies in their intuitive control. Moving the control points changes the curve's shape in predictable ways. The first control point affects the curve's initial direction, while the second control point influences the final direction. The curve always stays within the convex hull of its control points, making it stable and predictable for designers.
Bézier curves have revolutionized digital design and are fundamental to modern computer graphics. They are extensively used in vector graphics software, font design, animation systems, web development, and engineering applications. Their mathematical elegance combined with intuitive control makes them indispensable tools for creating smooth, scalable curves in virtually every field that involves digital design.