Bézier Curve History Definition Notes Examples Linear Bézier

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

1 History

2 Definition

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

P_i is called control point for the Bézier curve. A polygonA polygon (from the Greek poly for "many", and gonos for "angle") is a closed planar path composed of a finite number of sequential straight line segments. The straight line segments that make up the polygon are called its sides or edges and the points wh can be constructed by connecting the Bézier points with lineA line or straight line is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. a curve that is long and straight. Given two points, in Euclidean geometry, one can always find exactly one line that passes through thes, starting with P₀ and finishing with P_n. This polygon is called the Bézier polygon.

3 Notes

4 Examples

4.1 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

4.2 Quadratic Bézier curves

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

4.3 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:

5 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 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.

/****************************************************** Code to generate a cubic Bezier curve Warning - untested code *******************************************************/ 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 t, dt; int i; dt = 1.0 / ( numberOfPoints - 1 ); for( i = 0, t = 0; i < numberOfPoints; i++, t += dt) curve[i] = PointOnCubicBezier( cp, t ); }

6 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:

or simply

7 See also

8 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.

9 External links

Numerical analysis Curves

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