Dream & Passion: Bezier Curves

Linear Bézier curves

Given points P₀ and P₁, a linear Bézier curve is simply a straight line between those two points. The curve is given by

$\mathbf{B}(t)=\mathbf{P}_0 + t(\mathbf{P}_1-\mathbf{P}_0)=(1-t)\mathbf{P}_0 + t\mathbf{P}_1 \mbox{ , } t \in [0,1]$

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 P₀, P₁, and P₂,

$\mathbf{B}(t) = (1 - t)^{2}\mathbf{P}_0 + 2(1 - t)t\mathbf{P}_1 + t^{2}\mathbf{P}_2 \mbox{ , } t \in [0,1].$

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 P₀, P₁, P₂ and P₃ in the plane or in three-dimensional space define a cubic Bézier curve. The curve starts at P₀ going toward P₁ and arrives at P₃ coming from the direction of P₂. Usually, it will not pass through P₁ or P₂; these points are only there to provide directional information. The distance between P₀ and P₁ determines "how long" the curve moves into direction P₂ before turning towards P₃.

The parametric form of the curve is:

$\mathbf{B}(t)=(1-t)^3\mathbf{P}_0+3(1-t)^2t\mathbf{P}_1+3(1-t)t^2\mathbf{P}_2+t^3\mathbf{P}_3 \mbox{ , } t \in [0,1].$

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 farB(t) is from P₀ to P₁. For example when t=0.25, B(t) is one quarter of the way from pointP₀ to P₁. As t varies from 0 to 1, B(t) describes a curved line from P₀ to P₁.

[edit]Quadratic curves

For quadratic Bézier curves one can construct intermediate points Q₀ and Q₁ such that as t varies from 0 to 1:

Point Q₀ varies from P₀ to P₁ and describes a linear Bézier curve.
Point Q₁ varies from P₁ to P₂ and describes a linear Bézier curve.
Point B(t) varies from Q₀ to Q₁ 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 Q₀, Q₁ & Q₂ that describe linear Bézier curves, and points R₀ & R₁ 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 Q₀, Q₁, Q₂ & Q₃ that describe linear Bézier curves, points R₀, R₁ &R₂ that describe quadratic Bézier curves, and points S₀ & S₁ that describe cubic Bézier curves: