➰ Bezier Curves

Bezier curves are a type of mathematical curve that is widely used in computer graphics, computer-aided design (CAD), and other areas of computer science and engineering. They were first developed by Pierre Bézier, a French engineer, in the 1960s while working at the car manufacturer Renault to design curves for car bodies.

Definition

Bezier curves are defined by a set of control points that influence the shape of the curve. The simplest form of a Bezier curve is the quadratic Bezier curve, which is defined by three control points: the starting point, an intermediate control point, and the ending point. The curve starts at the starting point, approaches the intermediate control point, and ends at the ending point. The shape of the curve is determined by the positions of these control points.

Interactive Demo

Mathematics

The formula to calculate a quadratic Bezier curve between points ${\mathbf{P}}_0$ and ${\mathbf{P}}_2$, influenced by the control point ${\mathbf{P}}_1$, is given by:

$\mathbf{B}(t)=(1-t{)}^2{\mathbf{P}}_0+2(1-t)t{\mathbf{P}}_1+t^2{\mathbf{P}}_2$

Where:

• $\mathbf{B}(t)$ is the point on the curve at a time $t$ ($t$ ranges from 0 to 1).
• ${\mathbf{P}}_0$ is the starting point.
• ${\mathbf{P}}_1$ is the intermediate control point.
• ${\mathbf{P}}_2$ is the ending point.