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Bezier Curves Made Simple

, 14 Apr 2008 CPOL 186.9K 9.2K 89
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A simple implementation of the famous Bezier curves in C#. Easy to understand.



Bezier curves are the most fundamental curves, used generally in computer graphics and image processing. These curves are mainly used in interpolation, approximation, curve fitting, and object representation. In this article, I will demonstrate, in a very simple and straightforward way, how one can construct these curves and make use of them.


Bezier curves are parametric curves which are pretty much customizable and smooth. They are well suited for many applications. They were named after Pierre Bézier, a French mathematician and engineer who developed this method of computer drawing in the late 1960s while working for the car manufacturer Renault. People say that at the same time the same development took place during the research of Ford. There is still a confusion about who found it first.

Because of my imaging background, my article will mainly focus on interpolation and curve fitting. In interpolation, what one would simply like to do is to find unknown points using known values. This way, a discrete case can be represented with a more continuous structure, and we can have a well defined curve for missing points. The curve is initialized with certain data points, and it tries to generate new ones that are approximating (or interpolating) the old values.

Constructive Bezier Curve Algorithm

Consider the n+1 points P0,…,Pn and connect the points into a polyline we will denote hereafter as the control polygon.


Given points Pi, i = 0,...,n, our goal is to determine a curve g (t), for all values t Î [0,1]. The idea is demonstrated below:


Basic Algorithm

The objective here is to find points in the middle of two nearby points and iterate this until we have no more iterations. The new values of points will give us the curve. The famous Bezier equation is the exact formulation of this idea. Here is the algorithm:

Step 1: Select a value t Î [0,1]. This value remains constant for the rest of the steps.

Step 2: Set Pi[0] (t) = Pi, for i = 0,...,n.

Step 3: For j= 0,...,n, set simple.jpg for i = j,...,n.

Step 4: g (t) = Pn[n] (t)

Special & General Cases

Now, I will give formulas for common, special cases that can be helpful in certain applications. The code of the article does not demonstrate any of them, but it uses the generalized formula. So, let me start with the generalized formula:


For the sake of simplicity and convention used in this article and code, it is better to represent this formula as:


What this equation tells us is nothing but the formulation of the above algorithm (the mid-point iterations). It is very important in the sense that a whole algorithm could be summarized into a formula and a straightforward implementation would yield correct results. Here, n denotes the number of points and P denotes the points themselves. The factorial coefficients of the points are simply called the Bernstein basis functions, because of the name of the founder.

Here are the special cases:

Linear Bezier:


Quadratic Bezier:


Cubic Bezier:


Understanding and Using the Code

This is the function, doing all the work. I think it is very short and very easy. Because we are dealing only with 2D curves, we have points in X and Y coordinates. The function simply calculates the Bezier points.

public void Bezier2D(double[] b, int cpts, double[] p)
    int npts = (b.Length) / 2;
    int icount, jcount;
    double step, t;

    // Calculate points on curve

    icount = 0;
    t = 0;
    step = (double)1.0 / (cpts - 1);

    for (int i1 = 0; i1 != cpts; i1++)
        if ((1.0 - t) < 5e-6) 
            t = 1.0;

        jcount = 0;
        p[icount] = 0.0;
        p[icount + 1] = 0.0;
        for (int i = 0; i != npts; i++)
            double basis = Bernstein(npts - 1, i, t);
            p[icount] += basis * b[jcount];
            p[icount + 1] += basis * b[jcount + 1];
            jcount = jcount +2;

        icount += 2;
        t += step;

The rest of the functions are only helper functions taking part in factorial calculations and basis function calculations, which I believe are pretty obvious. To properly use this function, give it a set of points in the format: XYXYXYXYXYXYXYXYXYXY.... coordinates and how many points you would like to calculate on the curve. The function will fill the p array with the path points.

Points of Interest

Because of the limitations of factorial calculations, the code could only calculates curves up to 32 points. More complicated structures are generally represented by a combination of these curves (as in Adobe Photoshop, Illustrator, and Flash - path tool).

Even though GDI has a built-in Bezier curve calculation function, it is never good for learning to use built-in libraries. You won't always have GDI to do things for you! At some place, some time, you may have to implement it, and I think by now, you should have a rough idea of how these curves work.


This article, along with any associated source code and files, is licensed under The Code Project Open License (CPOL)


About the Author

Tolga Birdal
CEO Gravi Information Technologies and Consultancy Ltd
Turkey Turkey
Currently, also an MSc. student in Technical University of Munich, I develop practical application in computer vision for more than 5 years. I design real-time solutions to industrial and practical vision problems, both 3D and 2D. Very interested in developing algorithms in C relating math and vision.

Please visit Gravi's web page ( and my page ( to learn more about what we develop.

I admire performance in algorithms.

"Great minds never die, they just tend to infinity..."

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Comments and Discussions

QuestionGet bezier point list between two control points. Pin
Cyril Clavaud12-Aug-15 1:04
memberCyril Clavaud12-Aug-15 1:04 
AnswerRe: Get bezier point list between two control points. Pin
Tolga Birdal21-Sep-15 9:13
memberTolga Birdal21-Sep-15 9:13 
Question3D bezier curve Pin
Member 107473408-Apr-15 5:32
memberMember 107473408-Apr-15 5:32 
SuggestionPorted to C, FWIW Pin
Member 110512801-Sep-14 14:34
memberMember 110512801-Sep-14 14:34 
GeneralRe: Ported to C, FWIW Pin
Tolga Birdal1-Sep-14 14:46
memberTolga Birdal1-Sep-14 14:46 
GeneralRe: Ported to C, FWIW Pin
Member 110512804-Sep-14 5:48
memberMember 110512804-Sep-14 5:48 
GeneralExactly what I was looking for Pin
Member 795460812-Jan-13 12:12
memberMember 795460812-Jan-13 12:12 
GeneralRe: Exactly what I was looking for Pin
Tolga Birdal4-Aug-13 18:10
memberTolga Birdal4-Aug-13 18:10 
GeneralMy vote of 5 Pin
as4527-Nov-12 6:30
memberas4527-Nov-12 6:30 
many thanks for the detailed description and the useful advice
QuestionAnswer Pin
Member 912643914-Nov-12 9:06
memberMember 912643914-Nov-12 9:06 
AnswerRe: Answer Pin
Tolga Birdal14-Nov-12 10:10
memberTolga Birdal14-Nov-12 10:10 
SuggestionRe: Answer Pin
Member 167073211-Sep-13 18:23
memberMember 167073211-Sep-13 18:23 
GeneralRe: Answer Pin
Tolga Birdal11-Sep-13 21:17
memberTolga Birdal11-Sep-13 21:17 
GeneralMy vote of 5 Pin
manoj kumar choubey20-Feb-12 19:11
membermanoj kumar choubey20-Feb-12 19:11 
GeneralGreat work Pin
Rainer Halanek17-Mar-11 9:17
memberRainer Halanek17-Mar-11 9:17 
GeneralRe: Great work Pin
Tolga Birdal17-Mar-11 10:09
memberTolga Birdal17-Mar-11 10:09 
GeneralRe: Great work Pin
Rainer Halanek9-May-11 4:26
memberRainer Halanek9-May-11 4:26 
GeneralTebrik ederim Pin
Erol Esen8-Dec-10 3:54
memberErol Esen8-Dec-10 3:54 
GeneralRe: Tebrik ederim Pin
Tolga Birdal17-Mar-11 10:32
memberTolga Birdal17-Mar-11 10:32 
QuestionHow can i get smoother results? Pin
ifaris728-Nov-10 9:12
memberifaris728-Nov-10 9:12 
AnswerRe: How can i get smoother results? Pin
Tolga Birdal28-Nov-10 10:52
memberTolga Birdal28-Nov-10 10:52 
QuestionFilled closed bezier? Pin
Burisch7-Oct-10 0:01
memberBurisch7-Oct-10 0:01 
AnswerRe: Filled closed bezier? Pin
Sprat Fisher25-Nov-10 20:32
memberSprat Fisher25-Nov-10 20:32 
AnswerRe: Filled closed bezier? Pin
jeroen van dael21-Apr-11 6:08
memberjeroen van dael21-Apr-11 6:08 
GeneralMy vote of 5 Pin
yrodrigu25-Aug-10 5:39
memberyrodrigu25-Aug-10 5:39 

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