A **Bézier curve** is a parametric curve used in computer graphics and design to produce smooth curves and surfaces. These curves are described mathematically by polynomials and were widely popularized in the computer graphics industry due to their implementation in design and modeling programs.

The basic form of the **Bézier curve **uses four points:

**Starting point (P0)**: this is where the curve begins.**Control point 1 (P1)**: determines the direction and magnitude of the curvature at the start.**Control point 2 (P2)**: determines the direction and magnitude of the curvature at the end.**End point (P3)**: is where the curve ends.

The position of a point on the Bézier curve is calculated using a linear combination of these points and is usually defined by the **Bernstein cubic polynomial** formula. The exact shape of the curve is determined by the location of the control points.

## Characteristics and Advantages of Bézier Curves

The particular characteristics of Bézier curves offer a number of advantages that have led to their use in many industries, including graphic design:

**Precise control:**by manipulating the control points, users can easily adjust the shape of the Bézier curve to suit their needs.**Smoothness**: They ensure smoothness and continuity, making them ideal for representing contours and smooth shapes.**Modularity**: They can be chained or joined together to form longer or more complex paths, while maintaining continuity between segments.**Flexibility**: Bézier curves are available in different grades, from linear to higher grades, offering different levels of complexity and flexibility as needed.**Efficient interpolation**: Despite their flexibility, Bézier curves can be evaluated efficiently using algorithms such as De Casteljau.**Compact representation**: A Bézier curve is completely defined by its control points. This means that, regardless of its length or complexity, only a fixed number of points are required to represent it.**Affine invariance:**Bézier curves maintain their relative shape under affine transformations, such as translations, rotations and scaling.**Visibility of the envelope**: The control points form a convex envelope around the curve. This is useful because it guarantees that the curve will not leave this space bounded by the control points.**Intuitiveness**: For designers and artists, Bézier curves offer an intuitive way of working, since the movement or adjustment of a control point immediately shows its impact on the shape of the curve.**Widespread adoption**: Because of their usefulness and versatility, Bézier curves are supported by virtually all graphic design, 3D modeling, and related graphics and animation software.

## Types of Bézier Curves

There are different types of Bézier curves, and they are classified according to the number of control points they use. The complexity and flexibility of the curve increases with the number of control points. In the following we will describe the characteristics of the most common types:

### Linear (first degree) Bézier curves

- They use 2 control points: an initial point and an end point.
- It produces a straight line between these two points.
- The equation of a linear Bézier curve is simply a linear interpolation between two points.

### Quadratic Bézier curves (second degree)

- They use 3 control points: an initial point, a control point and an end point.
- They are useful for representing sections of parabolas.
- They are affected mainly by the position of the central control point.

### Cubic Bézier curves (third degree)

- They use 4 control points: a starting point, two control points and an end point.
- This type is the most commonly used in graphic design and animation, as it offers a good combination of control and smoothness.
- The two control points determine the tangency and shape of the curve at the start and at the end.

### Higher degree Bézier curves

- It is possible to define Bézier curves with more than 4 control points by increasing their degree.
- The higher the degree, the higher the complexity and flexibility of the curve.

However, in practice, Bézier curves with very high degrees can be difficult to control and may exhibit unwanted oscillations. For this reason, instead of using a very high degree curve, it is common to chain multiple lower degree Bézier curves together to form complex trajectories.

Although there are many types of **Bézier curves**, in many applications, especially in graphic design, cubic curves are the most commonly used because of their balance between control and complexity.

## What are Bézier Curves used for?

Thanks to their versatility and the level of control they offer, Bézier curves are used in a large number of areas, which we will describe below:

**Graphic design and typography**: In vector design programs such as Adobe Illustrator or CorelDRAW, Bézier curves are used to create and modify shapes. They are also fundamental in defining character outlines in digital fonts.**Animation and 3D modeling**: In Blender, Maya or 3ds Max, Bézier curves can be used to draw motion paths, create surfaces and define shapes.**Industrial design and CAD**: In computer-aided design, these curves can be used to accurately represent complex shapes and surfaces.**Automotive and aeronautical design**: Bézier curves are essential for the design of smooth and aerodynamic surfaces.**User interface**: In some graphical interfaces, transitions and animations can be based on Bézier curves to achieve more natural and fluid movements.**Video games**: Used for animations, object trajectories and surface modeling.**Simulations**: For tracing trajectories or movements in physics simulations or any other type of simulation that requires a smooth transition between points.**Video editing and visual effects**: For creating smooth camera movements, animations and transition effects.**Medical**: In medical visualization and in the creation of custom implants and prosthetics.**Music**: For interpolation of values in music editing and creation programs, such as in modulating effects or transitioning between notes.