Bézier 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, of which the Bézier triangle is a special case.

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.

Linear Bézier curves

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

TrueType fonts use Bézier splines composed of the quadratic Bézier curves.

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. In general, 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:


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

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

Generalization
The Bézier curve of degree $$n$$ can be generalized as follows. Given points P0, P1,..., Pn, the B&eacute;zier 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 $$n=5$$:
 * $$\mathbf{B}(t)=\mathbf{P}_0(1-t)^5+5\mathbf{P}_1t(1-t)^4+10\mathbf{P}_2t^2(1-t)^3+10\mathbf{P}_3t^3(1-t)^2+5\mathbf{P}_4t^4(1-t)+\mathbf{P}_5t^5 \mbox{, } t \in [0,1].$$

Terminology
Some terminology is associated with these parametric curves. We have
 * $$\mathbf{B}(t) = \sum_{i=0}^n \mathbf{P}_i\mathbf{b}_{i,n}(t),\quad t\in[0,1]$$

where the polynomials
 * $$\mathbf{b}_{i,n}(t) = {n\choose i} t^i (1-t)^{n-i},\quad i=0,\ldots n$$

are known as Bernstein basis polynomials of degree n, defining 00 = 1.

The points Pi are called control points for the B&eacute;zier curve. The polygon formed by connecting the B&eacute;zier points with lines, starting with P0 and finishing with Pn, that is, the convex hull of the Pi, is called the B&eacute;zier polygon, and the B&eacute;zier polygon contains the B&eacute;zier curve.

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. When more complex shapes are needed low order Bézier curves are patched together (obeying certain smoothness conditions) in the form of Bézier splines.

Code example
The following code is a simple practical example showing how to plot a cubic Bezier curve in the C programming language. 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 algorithms such as de Casteljau'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 de Casteljau's. The following code has been factored to make its operation clear - an optimization 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.

On some architectures, the code below can also be optimized by dynamic programming. E.g. since dt is constant, cx * t changes a constant amount with every iteration. By repeatedly applying this optimization, the loop can be rewritten without any multiplications (though such a procedure is not numerically stable).

/* Code to generate a cubic Bezier curve

typedef struct {   float x;    float y; } Point2D;

/* cp is a 4 element array where: cp[0] is the starting point, or A in the above diagram cp[1] is the first control point, or B cp[2] is the second control point, or C cp[3] is the end point, or D t is the parameter value, 0 <= t <= 1

Point2D PointOnCubicBezier( Point2D* cp, float t ) {   float   ax, bx, cx; float  ay, by, cy; float  tSquared, tCubed; Point2D result;

/* calculate the polynomial coefficients */

cx = 3.0 * (cp[1].x - cp[0].x); bx = 3.0 * (cp[2].x - cp[1].x) - cx; ax = cp[3].x - cp[0].x - cx - bx; cy = 3.0 * (cp[1].y - cp[0].y); by = 3.0 * (cp[2].y - cp[1].y) - cy; ay = cp[3].y - cp[0].y - cy - by; /* calculate the curve point at parameter value t */ tSquared = t * t;   tCubed = tSquared * t;    result.x = (ax * tCubed) + (bx * tSquared) + (cx * t) + cp[0].x;    result.y = (ay * tCubed) + (by * tSquared) + (cy * t) + cp[0].y;    return result; }

/* ComputeBezier fills an array of Point2D structs with the curve points generated from the control points cp. Caller must allocate sufficient memory for the result, which is 

void ComputeBezier( Point2D* cp, int numberOfPoints, Point2D* curve ) { float  dt; int	 i;

dt = 1.0 / ( numberOfPoints - 1 );

for( i = 0; i < numberOfPoints; i++) curve[i] = PointOnCubicBezier( cp, i*dt ); }

Another application for Bézier curves is to describe paths for the motion of objects in animations, etc. Here, the x, y positions of the curve are not used to plot the curve but to position a graphic. When used in this fashion, the distance between successive points can become important, and in general these are not spaced equally - points will cluster more tightly where the control points are close to each other, and spread more widely for more distantly positioned control points. If linear motion speed is required, further processing is needed to spread the resulting points evenly along the desired path.

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 Pi, the rational Bézier curve can be described by:

\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 } $$ or simply

\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 }. $$