4th Oct, 2018

Université Libre de Bruxelles

Question

Asked 30th Jul, 2014

A Bézier curve is a parametric curve frequently used in computer graphics, animation, modeling, CAD, CAGD, and many other related fields.

Bezier curves and surfaces are curves written in Bernstein basis form; so, they are known many years ago. However, these applications are used heavily only in the last 30 years. Why? What are your thoughts and opinions?

Bézier curves are also used in the time domain, particularly in animation and interface design, e.g., a Bézier curve can be used to specify the velocity over time of an object such as an icon moving from A to B, rather than simply moving at a fixed number of pixels per step. When animators or interface designers talk about the "physics" or "feel" of an operation, they may be referring to the particular Bézier curve used to control the velocity over time of the move in question.

The mathematical basis for Bézier curves — the Bernstein polynomial — has been known since 1912, but its applicability to graphics was understood half a century later. Bézier curves were widely publicized in 1962 by the French engineer Pierre Bézier, who used them to design automobile bodies at Renault. The study of these curves was however first developed in 1959 by mathematician Paul de Casteljau using de Casteljau's algorithm, a numerically stable method to evaluate Bézier curves, at Citroën, another French automaker.

Some more examples of the Bezier curves visualization:

- 134.08 KBGNU-12.jpg
- 104.79 KBGNU-Radiance_Bezier.jpg
- 140.45 KBGNU-09.jpg

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Dear @Abedallah, this thread is an example of beauty in mathematics! Yes! Share this question with @Demetris!

As You have already given plenty of information regarding Bezier curves, I am ready to propose some applications in some other fields than mentioned before! Good readings here at ResearchGate! "Bezier curves have become fundamental tools in many challenging and varied applications, ranging from computer- aided geometric design to generic object shape descriptors. Bezier curves have wide applications because they are easy to compute and very stable"!

Here are some examples in robotics,geology, economy...!

Conference Paper A control framework for snake robot locomotion based on shap...

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**Get help with your research**

Join ResearchGate to ask questions, get input, and advance your work.

Bézier curves are popular because their mathematical descriptions are compact, intuitive, and elegant. They are easy to compute, easy to use in higher dimensions (3D and up), and can be stitched together to represent any shape that you can imagine. These are extensively used in games to describe paths such as racing line, or flight paths. They are also used to describe font characters and automobile body shapes. They are also used for paths used in animation particularly for 3D animation. Most of these developments have grown in the past three decades.

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Dear @Abedallah, this thread is an example of beauty in mathematics! Yes! Share this question with @Demetris!

As You have already given plenty of information regarding Bezier curves, I am ready to propose some applications in some other fields than mentioned before! Good readings here at ResearchGate! "Bezier curves have become fundamental tools in many challenging and varied applications, ranging from computer- aided geometric design to generic object shape descriptors. Bezier curves have wide applications because they are easy to compute and very stable"!

Here are some examples in robotics,geology, economy...!

Conference Paper A control framework for snake robot locomotion based on shap...

11 Recommendations

Bézier curves are also popular in motion planning for another reason: they allow the specification of a path as a piece-wise continuous polynomial, thus avoiding (a) the problems of high-order polynomials (many inflection points), and (b) achieving continuously differentiable connections, which is not the case for, e.g., the equally famous approach to motion planning using Dubins paths.

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In this publication I used Bezier curves (in the form of Bernstein polynomials) to approximate solutions of non-local boundary problems.

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Dear @Nageswara, thank you very much for the detailed answer with very helpful information about the use of the Bezier curves in different disciplines; it is really amazing to read all of these applications to the Bezier curves.

In this chapter, J. Rossignac explained for potential user and developer of shape and animation design and visualization tools, what you you need to be familiar with the formulations, properties, limitations, and control techniques for such curves. Hence, in this chapter, he discusses a variety of curve representations and their use to create surfaces and animations.

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Dear @Ljubomir, thank you very much for the links that support a lot of information about the applications of the Bezier curves in different disciplines; yes indeed, it is a kind of beauty of Mathematics that I will share with dear @Demetris. The examples you supplied range from robotics, geology, economy, and macroscopic shaping.

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Yes indeed dear @Andreas, "Bézier curves are also popular in motion description for many other reasons: they allow the specification of a path as a piece-wise continuous polynomial, thus avoiding (a) the problems of high-order polynomials (many inflection points), and (b) achieving continuously differentiable connections, which is not the case for, e.g., the equally famous approach to motion planning using Dubins paths." The Bezier curves are built to be of Spline nature.

Dear @Ioannis, thank you for the article entitled: Polynomial approximation to a non-local boundary value problem by you and P. Palamides; in this article the Bezier curves are applied to find an approximation to the solution of non-local boundary problems in terms of Bernstein polynomial basis. I give the abstract for the paper: a special class of polynomials which converge uniformly to the solution of a non-local boundary value problem (NBVP) is constructed. The use of this special class is justified by the physics of the model which is described by this NBVP. This NBVP has been studied by Palamides et al. in 2009, where the existence of solutions is established.

It seems that the Bezier curve is a result of a dynamic Interpolation made recursively. If one gives two points, the natural dynamic Interpolation is the segment parametrized (1-t)P0 + t P1. Now, if parametric curve A is defined for points P0,... , P_(n-1) and parametric curve B is defined for points P1, ..., P_n; then one defines for points P0, ..., P_n the curve given by C(t) = (1-t)A(t) + tB(t). So a Bezier curve for n points is a transverval of the homotopic family of curves connecting two Bezier curves for n-1 points. So finally the Bezier curve is a very smooth approximation of a polygonal line (sequence of segments P0P1, P1P2, ..., P_(n-1)P_n) that dynamically processes the very unsmooth tendencies of the polygonal line. One calls this poygonal line "control polygon". One can move the vertices of the control polygon since the Bezier curve has the form one whishes. This is very practical. Another application is given by related curves doing just Interpolation - called incidentally also Bezier curves. One looks for the polynomial curve passing through given n points. This is also very practical, for example for the users of the primary graphic abilities of LATEX. My first parabola in Latex was in fact an interpolation Bezier going though four given points!

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Dear @Mihai, thank you for your contribution; the process you mentioned is a linear interpolation which is the easiest method; it is called the de Casteljau algorithm which very practical method to draw a curve on the computer. The other application you just mentioned is drawing any pdf file using the cubic Bezier curves.

We have used some types of drawing gear at our undergraduate schools to produce engineering drawings - protractor, compass, drafter to name a few. The compass and drafter have control points and we are well aware of the flexibility that they offer in producing a sketch. In addition I hope you will all agree on the intuitive "French Curves" ! They can be exploited to produce a wide range of curves. Possibly these handy tools were recreated as Bezier curves in this present digital age. I am trying to correlate the same kind of jobs performed earlier on a drawing sheet and today in computers. Bezier might be nostalgic with the French curves, that he has put them in computers !

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Thanks for sharing your experience dear @Arun in producing engineering drawings - protractor, compass, drafter using some types of drawing gear. The use of control points facilitate flexibility in producing a sketch. Possibly these handy tools were recreated as Bezier curves in this present digital age.

I give some information about the French curves:

A French curve is a template made out of metal, wood or plastic composed of many different curves. It is used in manual drafting to draw smooth curves of varying radii. The shapes are segments of the Euler spiral or clothoid curve.

The curve is placed on the drawing material, and a pencil, knife or other implement is traced around its curves to produce the desired result.

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I think their flexibility due to the property of locating one point while drawing the other points, very-very roughly speaking, of course. Theoretically, probably due to the stabler nature of Bernstein polynomials compared to other polynomial sets.

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Thank you dear @Demetris for joining the discussion. Yes indeed flexibility is one of the important features of Bezier curves besides other properties including the stability based on the of Bernstein polynomial basis..

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Deleted profile

The great advantage of the Bezier-technique is that, via the control points, you can very well estimate how the curve will change if you exchange one of the control points (convex cover). On the other hand, while plotting a Bezier-curve, in fact only the control-points which are won by repeatingly sub-dividing the Bezier-net are used (compare the de Casteljau-algorithm).

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In addition to what has already been mentioned about Bezier curves, it has been observed that a Bezier curve B(t) always passes through the first and the last control point and lies within the convex hull of its control points (see attached image). See, for example, the proof of Lemma 10, p. 18, in

B. Pokorna, Quadratic space-like Bezier curves in three dimensional Minkowski space, 2012:

The important thing to notice is that by locating Bezier curves in convex hulls, we gain access a number of useful views of Bezier curves in real vector spaces and bring into the picture results obtained in geometry, algebra and topology.

- 49.66 KBBezierConvexHull.jpg

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Bézier splines were used extensively in CATIA v4 and other CAD-systems. The reason for this is that they produce very smooth curves with an excellent C1 and C2 continuity which makes them ideal for design purposes. However because of the fact the whole Bézier spline behaves as a spline with one segment (where a Bézier with n points creates a polynom of degree n-1) it is extremely difficult to get a good grip on the shape of the curve while designing. This is one of the main reasons they have now been replaced in modern CAD/CAM-systems by NURBS.

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Dear @Katja, thank you for the contribution. The control points determine the shape of the

Bezier curve, and, while plotting a Bezier curve, only the de Casteljau control-points are used; therefore simplifying the calculations.

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Dear @James, Thank you for joining the forum. Thank you for the graph and the interesting article. A Bezier curve passes through the first and the last control points and lies within the convex hull of its control points. One important thing to notice is that by locating Bezier curves in convex hulls, we gain access a number of useful views of Bezier curves in real vector spaces and bring into the picture results obtained in geometry, algebra and topology. yes indeed, this is very important in determining a proper space for robots.

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Yes indeed dear @John, Almost all CAD systems including CATIA v4 do use Bézier splines. The reason for this is that they produce very smooth curves with an excellent C1 and C2 continuity which makes them ideal for design purposes.Well, if the resulting spline is not satisfactory, then other kinds including rational forms can be used.

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Dear @Ravi Pottathil sent me his opinion and shared the following article the explains some facts about the Bezier curves: "Excellent answers are already given by above experts. In addition what made this really popular is the easy computer coding schemes.

see a simple article by Toga Bridal on this subject

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Dear @G. Indika P. Perera, B-Spline has advantage that remove/add one control point does not change the whole curve but only local changes.

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Dear Indika

B-splines: In the case of Bézier curve, it is considered as a single curve controlled by all the control points. As a result, with an increase in the number of control points, the order of the polynomial representing the curve increases. To reduce this complexity, the curve is broken down into more segments with better control exercised with individual segments, while maintaining a simple continuity between the segments. An alternative is to use a B-spline to generate a single piecewise parametric polynomial curve through any number of control points with the degree of the polynomial selected by the designer. B-splines exhibit a local control in that whenever a single vertex is moved, only those vertices around that will be affected while the rest remains the same.

The B-spline surface thus combines the strengths of Coons and Bézier surface definitions by forming the surface from a number of controlling curves similar to Coon’s while having the control points to alter surface curvature. This is the main reason all CAD systems use B-splines and NURBS. Smoothing of surfaces particularly for sculptured surfaces can be done selectively by tweaking individual points.

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A Bézier curve can be used to specify the velocity over time of an object such as an icon moving from A to B, rather than simply moving at a fixed number of pixels per step. however, original data can be lost by smoothing technique.

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Dear @Nageswara, thank you for the active participation. Dear @Hashem, thank you for your contribution, Bezier curve passes through the first and last points.

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I think that the importance of Bezier curves, steams from their connection with computational algebraic geometry. It is well known that Algebraic geometry have a lot of applications starting from simple things to robotics. This is I think the main reason, see for example:

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Dear @Costas, thank you for the article that links the quadratic Bezier curves to computational algebraic geometry, groebner basis, and rational implicitization.

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Dear @Arno, , thank you for this the information on the use of Bezier curves in theory of the origin of time in Cantor's universe; it is really interesting.

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Dear @Georges Fadel sent me his opinion that I share with you: "easy to program, ability to control shape by moving control points. It was developed in France by Pierre Bezier for CAD representation. It does not allow local control, so moving one control point would affect the whole curve, mostly around the point that is moved. But it is an easy extension of parametric cubic representation, much easier to understand than B-splines and NURBS."

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First of all, Bézier curves were invented because they were necessary (hence they were developed by two people independently at around the same time). The Bernstein polynomials have the really important mathematical property that the basis functions add to one everywhere. I think they are still so popular because of their intuitive character. I have used Bézier curves long before I could have even understood the mathematics behind them (i.e. as a child). Having only four control points makes it easy to control the behaviour of the curve. Other splines are of no use as long as they are not easy to manipulate. (Easy also means that you don't need to understand the mathematics in order to properly make use of it.) Therefore I guess that we will never find a suitable replacement for Bézier curves.

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Dear @Simon, thank you for sharing and making good points: you are right using the Bézier curves long before I could have even understood the mathematics behind them (i.e. as a child) means that their use is simple and straightforward. Having only four control points makes it easy to control the behaviour of the cubic curve and, therefore, it will never be easy to find a suitable replacement for Bézier curves.

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Another aspect of Bezier curves that makes them attractive is their extension of B-spline curves. For example, B-spline curves can be used to extract convex sets from the vertices of the convex polygons in a Voronoi diagram. The advantage in doing this is both artistic and practical. From an artistic point of view, the rigid convex polygons in a Voronoid diagram can be a source beautiful nonlinear shapes that are convex sets with curved boundaries that contain some of the vertices in the Voronoi polygons. From a practical point of view, the nonlinear convex shapes extractable from a Voronoi diagram can be used in solving shape recognition problems in digital images.

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Dear @James, thank you for the repeated valuable contribution on this thread; it is highly appreciated. The link between the Bezier and the B-splines is a very important issue that has both theoretical and practical uses that has many uses in design and modeling.

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It is true dear @James and @Abedallah. "B-spline curves share many important properties with Bézier curves, because the former is a generalization of the later. "

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Dear @Ljubomir, thank you for the repeated appreciated contributions on this thread; this article contains many important properties of the B-Splines. A B-spline curve C(u) of degree p is defined by n + 1 control points and a knot vector U = { u0, u1, ...., um } with the first p+1 and last p+1 knots "clamped" (i.e., u0 = u1 = ... = up and um-p = um-p+1 = ... = um)..

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The Bezier curves are much smoother, more softly generalized and gently looking comparing to splines. Just comparing these two pictures from my MSc Thesis: the same data, just two various approaches - Bezier and splines:

- 78.35 KBGNU-Irr.jpg
- 81.16 KBGNU-Irr-Splines.jpg

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Scientists Support Ukraine

Discussion

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- Asked 3rd Mar, 2022

- Ijad Madisch

Like so many, I am shocked and saddened at seeing war break out in Europe. My thoughts – and those of the ResearchGate team – are with the people of Ukraine and everyone affected.

ResearchGate is an international company, whose purpose is to enable scientists across the world to work together openly and collaboratively, regardless of borders or nationality. We have people from over 40 countries on our staff of around 200, and being based in Berlin, we are profoundly aware of the human cost of conflicts, the echoes of which have shaped and scarred our home city. We join with the international community in condemning the actions of the Russian state.

From today, we will offer free advertising space worth $2.5 million on our network to humanitarian organizations working to respond to the crisis. ResearchGate benefits from over 50 million visitors every month, and we hope this initiative can help raise funds and awareness for those organizations that are having direct impact and need support.

We also want to use our platform to highlight the response from the scientific community. Personally, I have found the messages of support from scientists everywhere to be truly heartfelt, and I would like to highlight some of the community initiatives I’ve seen here:

**Science for Ukraine**provides an overview of labs offering a place for researchers and students who are affected to work from, as well as offers of employment, funding, and accommodation: https://scienceforukraine.eu/**Labs supporting Ukrainian Scientists**is an expansive list of labs and PIs offering support at this time. Submissions go here: https://docs.google.com/forms/d/e/1FAIpQLSeRGe5Da_b6GGyC6VT7CLGViGs06SzeuX7wRKpC4K5tnvlhgg/viewform Find the full list here: https://docs.google.com/spreadsheets/d/1HqTKukfJGpmowQnSh4CoFn3T6HXcNS1T1pK-Xx9CknQ/edit#gid=320641758- This
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**Doctors Without Borders / Médecins Sans Frontières (MSF)**: https://www.doctorswithoutborders.org/what-we-do/countries/ukraine**The UN Refugee Agency**: https://donate.unhcr.org/int/en/ukraine-emergency**UNICEF**: https://www.unicef.org/emergencies/conflict-ukraine-pose-immediate-threat-children

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This list outlines country-level initiatives from various academic institutions and research organizations, with a focus on programs and sponsorship for Ukrainian researchers:

Article

- Jan 2021

This article reveals a mathematical bridge between compass-only geometric constructions and parametric curves. A set of construction algorithms are presented which are used to locate points on any Bézier curve or B-Spline by using an abstract compass. These constructions aim to translate the B-Spline definition of Paul de Casteljau into the modifie...

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