A Bézier curve is a mathematical curve used in geometry and computer graphics to model smooth, flexible shapes. It was invented by Pierre Bézier, an engineer at Renault, to design car bodies. Today, Bézier curves are widely used in fields such as graphic design, animation, vector graphics and software development.

Characteristics of Bézier curves

1. Control points:
   • A Bézier curve is defined by a set of control points. The curve itself does not necessarily pass through all these points, except for the extreme points (the first and last).
   • These points influence the shape of the curve. Moving the control points modifies the curve interactively, allowing precise control of the shape.
2. Bézier curve types:
   • Linear Bézier curve: The simplest form, defined by two points. It’s simply a straight line between these two points.
   • Quadratic Bézier curve: Defined by three control points. The curve is influenced by a single intermediate control point, which draws the curve towards it.
   • Cubic Bézier curve: Defined by four control points. This is the most common shape used in graphics software. It can be used to create more complex, smoother curves.
3. Interpolations:
   • Bézier curves are generated by interpolating control points. This means that the curve is constructed by calculating intermediate points between the control points according to a certain proportion.
4. Mathematical equation:
   • Bézier curves can be expressed mathematically using polynomial formulae. For example, a cubic Bézier curve is represented by a linear combination of the four control points, weighted by coefficients based on parametrization.

Bézier curve applications

1. Vector graphics:
   • Bézier curves are at the heart of vector graphics software such as Adobe Illustrator, CorelDRAW and Inkscape. They enable the creation of complex shapes with smooth curves that can be resized without loss of quality.
2. Animation:
   • In 2D and 3D animation, Bézier curves are used to define object trajectories or to control the timing of animations. They help create fluid, natural movements.
3. Typography:
   • Bézier curves are also used to define letter outlines in fonts. This makes it possible to create fonts with precise, adaptable shapes.
4. Web development:
   • In CSS (Cascading Style Sheets), Bézier curves are used to define fluid animations and transitions. For example, the cubic-bezier() function lets you customize the transition curve for web animation effects.

Bézier curves are powerful tools for modeling smooth, controllable shapes in many graphic and technical disciplines. Their flexibility and mathematical simplicity make them an essential choice for any situation requiring smooth, precise curves.