Bézier curve

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Cubic Bézier curve

In the mathematical field of numerical analysis, a Bézier curve is a parametric curve important in computer graphics. Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangleis a special case.

Bézier curves were widely publicised in 1962 by the French engineer Pierre Bézier, who used them to designautomobile bodies. The curves were first developed in 1959 by Paul de Casteljau using de Casteljau's algorithm, anumerically stable method to evaluate Bézier curves.

[edit] Examination of cases

[edit] Linear Bézier curves

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

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

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,

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

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:

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

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

[edit] Generalization

The Bézier curve of degree

n

can be generalized as follows. Given points P₀, P₁,..., P_n, the Bézier curve is

$\mathbf{B}(t)=\sum_{i=0}^n {n\choose i}\mathbf{P}_i(1-t)^{n-i}t^i =\mathbf{P}_0(1-t)^n+{n\choose 1}\mathbf{P}_1(1-t)^{n-1}t+\cdots+\mathbf{P}_nt^n \mbox{ , } t \in [0,1].$

For example, for

n = 5

$\mathbf{B}(t)=\mathbf{P}_0(1-t)^5+5\mathbf{P}_1t(1-t)^4+10\mathbf{P}_2t^2(1-t)^3+10\mathbf{P}_3t^3(1-t)^2+5\mathbf{P}_4t^4(1-t)+\mathbf{P}_5t^5 \mbox{ , } t \in [0,1].$

This formula can be expressed recursively as follows: Let $\mathbf{B}_{\mathbf{P}_0\mathbf{P}_1\ldots\mathbf{P}_n}$ denote the Bézier curve determined by the points P₀, P₁,..., P_n. Then

$\mathbf{B}(t) = \mathbf{B}_{\mathbf{P}_0\mathbf{P}_1\ldots\mathbf{P}_n}(t) = (1-t)\mathbf{B}_{\mathbf{P}_0\mathbf{P}_1\ldots\mathbf{P}_{n-1}}(t) + t\mathbf{B}_{\mathbf{P}_1\mathbf{P}_2\ldots\mathbf{P}_n}(t)$

In words, the degree

n

Bézier curve is a linear interpolation between two degree

n - 1

Bézier curves.

图例示意

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 quadratic Bézier curves, and points R₀ & R₁ that describe linear 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 cubic Bézier curves, points R₀, R₁ & R₂ that describe quadratic Bézier curves, and points S₀ & S₁ that describe linear Bézier curves:


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

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