Bezier curve

In the mathematical subfield of numerical analysis a Bézier curve is a parametric curve important in computer graphics. A numerically stable method to evaluate Bézier curves is de Casteljau's algorithm.

Generalizations of Bézier curves to higher dimensions are called Bézier surfaces; the Bézier triangle is a special case.

Bézier curves are also formed by many common forms of string art, where strings are looped across a frame of nails.

History

Bézier curves were widely publicized in 1962 by the French engineer Pierre Bézier who used them to design automobile bodies. The curves were developed in 1959 by Paul de Casteljau using de Casteljau's algorithm.

Definition

Given n+1 points P_i in R³ a Bézier curve of degree n is a parametric curve

<math>\mathbf{B}:[0,1] \to \mathbb{R}^3<math>

composed of Bernstein basis polynomials of degree n

<math>\mathbf{B}(t)= \sum_{i=0}^n \mathbf{P}_{i} b_{i,n}(t) \mbox{ , } t \in [0,1]<math>

with the Bernstein basis polynomials defined as

<math>b_{i,n}(t):= {n \choose i} t^i (1-t)^{n-i} \qquad \mbox{ , } i=0,\ldots n<math>

P_i is called control point for the Bézier curve. A polygon can be constructed by connecting the Bézier points with lines, starting with P₀ and finishing with P_n</sub>. This polygon is called the Bézier polygon.

Notes

• The starting point of the curve is P₀ and the ending point is P_n
• The Bézier curve is completely contained in the convex hull of the control points.
• If and only if all control points lie on the curve it is a straight line.
• The start (end) of the curve is tangent to the first (last) section of the Bézier polygon.
• A curve can be split into arbitrarily many subcurves, each of which is also a Bézier curve.
• A Bézier curve cannot form a true circle.
• A second Bézier curve that is truly parallel to an existing curve cannot be derived mathematically from the control points of the first (though there are heuristic methods that work for practical purposes).

Examples

Linear Bézier curves

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

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

Quadratic Bézier curves

A quadratic Bézier curve is the path traced by the function B(t). For points A, B, and C,

<math>\mathbf{B}(t) = (1 - t)^{2}\mathbf{A} + 2t(1 - t)\mathbf{B} + t^{2}\mathbf{C} \mbox{ , } t \in [0,1]<math>

Cubic Bézier curves

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

The parametric form of the curve is:

<math>\mathbf{B}(t)=\mathbf{A}(1-t)^3+3\mathbf{B}t(1-t)^2+3\mathbf{C}t^2(1-t)+\mathbf{D}t^3 \mbox{ , } t \in [0,1]<math>.

Application in computer graphics

Bézier curves are widely used in computer graphics to model smooth curves. As the curve is completely contained in the convex hull of its control points, the points can be graphically displayed and used to manipulate the curve intuitively. Affine transformations such as translation, scaling and rotation can be applied on the curve by applying the respective transform on the control points of the curve.

The most important Bézier curves are quadratic and cubic curves. Higher degree curves are more expensive to evaluate and there is no analytic formula to calculate the roots of polynomials of degree 5 and higher. When more complex shapes are needed low order Bézier curves are patched together (obeying certain smoothness conditions) in the form of Bézier splines. Modern imaging systems like PostScript, Metafont and GIMP use Bézier splines composed of cubic Bézier curves for drawing curved shapes. TrueType fonts use Bézier splines composed of the simpler quadratic Bézier curves.

The following code is a simple practical example showing how to plot a cubic Bezier curve in C. Note, this simply computes the coefficients of the polynomial and runs through a series of t values from 0 to 1 - in practice this is how it is usually done, even though neat algorithms such as deCasteljau's are often cited in graphics discussions, etc. This is because in practice a linear algorithm like this is faster and less resource-intensive than a recursive one like deCasteljau's. The following code has been factored to make its operation clear - an optimisation in practice would be to compute the coefficients once and then re-use the result for the actual loop that computes the curve points - here they are recomputed every time, which is less efficient but helps to clarify the code.

The resulting curve can be plotted by drawing lines between successive points in the curve array - the more points, the smoother the curve.

Rational Bézier curves

Some curves that seem simple, like the circle, cannot be described by a Bézier curve or a piecewise Bézier curve (though in practice the difference is small and may be tolerable). To describe some of these other curves, we need additional degrees of freedom.

The rational Bézier curve adds weights that can be adjusted. The numerator is a weighted Bernstein form Bézier curve and the denominator is a weighted sum of Bernstein polynomials.

Given n+1 control points P_i, the rational Bézier curve can be described by:

<math>

\mathbf{B}(t) = \frac{ \sum_{i=0}^n b_{i,n}(t) \mathbf{P}_{i}w_i } { \sum_{i=0}^n b_{i,n}(t) w_i } <math> or simply

<math>

\mathbf{B}(t) = \frac{ \sum_{i=0}^n {n \choose i} t^i (1-t)^{n-i}\mathbf{P}_{i}w_i } { \sum_{i=0}^n {n \choose i} t^i (1-t)^{n-i}w_i } <math>

See also

• Spline (mathematics)
• Bézier spline
• Bézier surface
• Bézier triangle
• NURBS

References

• Paul Bourke: Bézier curves, http://astronomy.swin.edu.au/pbourke/curves/bezier/
• Donald Knuth: Metafont: the Program, Addison-Wesley 1986, pp. 123-131. Excellent discussion of implementation details; available for free as part of the TeX distribution.