##### Linear Bézier curves

Given points **P**_{0} and **P**_{1}, a linear Bézier curve is simply a straight line between those two points. The curve is given by

and is equivalent to linear interpolation.

##### [edit]Quadratic Bézier curves

A quadratic Bézier curve is the path traced by the function **B**(*t*), given points **P**_{0}, **P**_{1}, and **P**_{2},

A quadratic Bézier curve is also a parabolic segment.

TrueType fonts use Bézier splines composed of quadratic Bézier curves.

##### [edit]Cubic Bézier curves

Four points **P**_{0}, **P**_{1}, **P**_{2} and **P**_{3} in the plane or in three-dimensional space define a cubic Bézier curve. The curve starts at **P**_{0} going toward **P**_{1} and arrives at **P**_{3} coming from the direction of **P**_{2}. Usually, it will not pass through **P**_{1} or **P**_{2}; these points are only there to provide directional information. The distance between **P**_{0} and **P**_{1} determines "how long" the curve moves into direction **P**_{2} before turning towards **P**_{3}.

The parametric form of the curve is:

Modern imaging systems like PostScript, Asymptote and Metafont use Bézier splines composed of cubic Bézier curves for drawing curved shapes.

#### Constructing Bézier curves

##### [edit]Linear curves

Animation of a linear Bézier curve, *t* in [0,1]

The *t* in the function for a linear Bézier curve can be thought of as describing how far**B**(*t*) is from **P**_{0} to **P**_{1}. For example when *t=0.25*, **B**(*t*) is one quarter of the way from point**P**_{0} to **P**_{1}. As *t* varies from 0 to 1, **B**(*t*) describes a curved line from **P**_{0} to **P**_{1}.

##### [edit]Quadratic curves

For quadratic Bézier curves one can construct intermediate points **Q**_{0} and **Q**_{1} such that as *t* varies from 0 to 1:

- Point
**Q**_{0}varies from**P**_{0}to**P**_{1}and describes a linear Bézier curve. - Point
**Q**_{1}varies from**P**_{1}to**P**_{2}and describes a linear Bézier curve. - Point
**B**(*t*) varies from**Q**_{0}to**Q**_{1}and describes a quadratic Bézier curve.

Construction of a quadratic Bézier curve

Animation of a quadratic Bézier curve, *t* in [0,1]

##### [edit]Higher-order curves

For higher-order curves one needs correspondingly more intermediate points. For cubic curves one can construct intermediate points **Q**_{0}, **Q**_{1} & **Q**_{2} that describe linear Bézier curves, and points **R**_{0} & **R**_{1} that describe quadratic Bézier curves:

Construction of a cubic Bézier curve

Animation of a cubic Bézier curve, *t* in [0,1]

For fourth-order curves one can construct intermediate points **Q**_{0}, **Q**_{1}, **Q**_{2} & **Q**_{3} that describe linear Bézier curves, points **R**_{0}, **R**_{1} &**R**_{2} that describe quadratic Bézier curves, and points **S**_{0} & **S**_{1} that describe cubic Bézier curves:

Construction of a quartic Bézier curve

Animation of a quartic Bézier curve, *t* in [0,1]

(See also a construction of a fifth-order Bezier curve.)

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