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What is G1 continuity? --AxelBoldt
Continuous curves with tangents pointing in the same direction. Used by graphics/font workers http://www.google.com/search?q=g1.continuity
What is the point on the cubic curve in the recursive algorithm?
What does "A truly parallel Bézier curve cannot be derived mathematically" mean?
Wow, if only there was an explanation for the layman. Someone below claims this is it: I did maths to the end of school and have no idea about this, and in particular the use of these curves in Illustrator remains a mystery to me. Please help!
I want to merge Bézier curve and Bernstein polynomial. I do not care what the resulting article is called. The two article are talking about the same subject and so there is a lot of duplicated material and inconsistent notation. Any comments ? MathMartin 12:52, 19 Sep 2004 (UTC)
I changed my opinion. The material should stay on separate pages, although the notation should be more consistent. What is the term for splines that are patched together using polynomials in Bézier form ? And what exactly is a Bezigon ? Is the term common ? What I was trying to do when rewriting Bézier curve and Bernstein polynomial was to make the connection between the two topics clearer. At the moment I think the articles should be structured like this
It is not clear to me what the last two pages should be called, perhaps someone with more knowledge can propose some naming. MathMartin 10:20, 20 Sep 2004 (UTC)
Done. Currently there are 3 articles exactly as you suggested. The article about "a spline composed of polynomials in Bezier form" is titled Bézier spline, with Bezigon as a redirect to that article, which I think is appropriate. -- DavidCary ( talk) 04:48, 21 February 2013 (UTC)
I removed the following two paragraphs from the page. The first does not contain information, and the second is a bit obscure. MathMartin 17:09, 19 Sep 2004 (UTC)
You misunderstood me, perhaps because my comment was too cryptic. I understand the sentence
I put the mathematical definition at the top, so there is no need to generalize form the cubic case to higher order curves. Stating that only low order curves are used in the industry without any reason is not very useful.
Now that is a good reason not to use high order Bezier curves, you should have put it in the article. I intend to expand the article in the next few days and will include it. Just a short correction, even roots of a fifth order polynomial can generally not be found analytically.
I understand the need to patch together low degree Bezier curves to form splines, and that one must observe certain smoothness conditions when patching the curves together. But I think Bezier curve article is the wrong place to talk about splines, (to some extend I already talked about splines in the introduction) and the paragraph was very badly written. What is the direction C-D ? What is G1 continuity ? You do not have to explain those terms to me, I know them or can guess them. But for the reader who does not know the material, who just wants to look up some information, the paragraph is obscure.
To sum up the gist of my arguments. Yes I am a mathematician. I believe there should be formal mathematical definition even if they scare the laymen. But there should be real world explanations and examples too. But the math definitions and the laymen explanations should not be mixed, they should be put in separate sections. In an encylopaedia it is better to provide a general definition and then give examples then to give an example and then generalize form this example.
On a different note perhaps you can shed some light on the notation used on the page. What is the deal with the upper case Bézier coefficients (A, B etc.) written in bold ? Why not call them p_0, p_1 ..? MathMartin 10:20, 20 Sep 2004 (UTC)
Points on a quadratic Bézier curve can be computed recursively:
The points on a Bézier curve can be quickly computed using a recursive procedure which uses division by two as its fundamental operation and avoids floating point arithmetic altogether;
D:=(C+D)/2, C:=(B+C)/2, D:=(C+D)/2, B:=(A+B)/2, C:=(B+C)/2, D:=(C+D)/2 (No language in particular)
I removed the above paragraphs from the article. They don't make any sense to me. MathMartin 15:00, 20 Sep 2004 (UTC)
Is it possible to return structures in C? (Haven't tried testing the code.)
Here's something like what I'd write (which isn't quite as long), to do the same thing. Haven't tested. Not sure whether 2 multiplies or 6 add/subtracts are faster.
#define ComputeBezier(a,b,c) computebezier((float *)(a),(b),(float *)(c)) void computebezier(float *cp, int n, float *res) { float s, s2, s3, t, t2, t3, st2, s2t, ds; float ax, ay, bx, by, cx, cy, dx, dy; ax=*cp++; ay=*cp++; bx=3**cp++; by=3**cp++; cx=3**cp++; cy=3**cp++; dx=*cp++; dy=*cp; ds=1./(n-1); s=0; t=1; while(n--) { s+=ds; s2=s*s; s3=s*s2; t-=ds; t2=t*t; t3=t*t2; st2=s*t2; s2t=s2*t; *res++=ax*t3+bx*st2+cx*s2t+dx*s3; *res++=ay*t3+by*st2+cy*s2t+dy*s3; } }
Κσυπ Cyp 11:03, 21 Sep 2004 (UTC)
private void bezier(Graphics g, int n, double x1, double y1, double x2, double y2, double x3, double y3){ double sx1=(x2-x1)/n, sx2=(x3-x2)/n, sy1=(y2-y1)/n, sy2=(y3-y2)/n; for(int i=0;i<n;++i){ g.drawLine((int)(x1+i*sx1+i*i*(sx2+sx1)/n), (int)(y1+i*sy1*2+i*i*(sy2-sy1)/n), (int)(x1+(i+1)*sx1+(i+1)*(i+1)*(sx2+sx1)/n), (int)(y1+(i+1)*sy1*2+(i+1)*(i+1)*(sy2-sy1)/n)); } }
Just a short note on the introduction.
Generalizations of Bézier curves to three (or more) dimensions are called a Bézier surface and a Bézier triangle.
This is not a very clear formulation. In mathematics a curve is a 1 dimensional object because it depends on one parameter t. A Bézier surface would be a 2 dimensional object because it depends on 2 parameters. Consider this explanation: a curve can be drawn in a 2 dimensional space or in a 3 dimensional space (or any dimension for this matter) but it still is a curve, a 1 dimensional object.
For me this is clear. But other people may not know this. Perhaps someone can reformulate the sentence to make it clearer. Until then I will just remove the exact dimension number and speak about higher dimension. MathMartin 12:22, 21 Sep 2004 (UTC)
Mathematically the bezier curve may be one dimensional, but intuitively it's two dimensional because it produces a point (x,y).
I rewrote the defininition to clear this up. How many points the curve produces is dependend on the dimension of the control points. If I choose control points in 3 dimensional space I get a curve in 3 dimensional space. If this is not clear I will expand the material in the article.
mathematicians (who will have better references to work from than this)
You should not make assumptions about the audience :) Even mathematicians need to look stuff up, and start from the beginning when learning a new topic. Most math books are crap because they do not contain enough examples and motivation.
An encyclopaedia should be accessible to a wide range of people. I think at the moment the article is well balanced. The laymen can read and understand the history section (should be expanded) and the applications in computer graphics sections. The programmer can grab the examples and implement them. And the mathematician (or any person with some training in math) can lookup the definition.
I believe it's easier to relate the code to the maths in the article than your example, which is no doubt faster... we must remember that this is an encyclopaedia, not a code text book - maybe even C is too technical here, a BASIC (shudder) or Pascal example might be better for readability
I agree. The code should match the math in the article and perhaps C is not the best choice. But as far as I know there is no standard computer language on wikipedia for examples, so whoever writes the code gets to choose the language. In my opinion code should only be used the make algorithms clearer. Complete implementations should be put at http://wikisource.org/wiki/Wikisource:Source_code and linked form the article.
MathMartin 12:11, 22 Sep 2004 (UTC)
should be put here http://wikisource.org/?title=Beizer_Curve&action=edit preferred language should be Python cause it's a high level, easily readable, used by mathematicians worldwide language that has bright future (pascal/fortran/basic are dead)
The coefficients for the cubic curive are calculated wrong in the example source code given. This can easily be verified by drawing lines between the control points and see that the points on the curve calculated by the example code goes outside the lines between the control points.
The correct coefficients should be (easily expanded from the parametric form):
such as:
The code for generating this (in Java) is:
private Point pointOnCubic(float t) { Point res = new Point(); float t2 = t * t; float t3 = t2 * t; // cp[] is an array of Point's specifying the control points float ax = -cp[0].x + 3*cp[1].x - 3*cp[2].x + cp[3].x; float bx = 3*cp[0].x - 6*cp[1].x + 3*cp[2].x; float cx = 3*cp[1].x - 3*cp[0].x; float ay = -cp[0].y + 3*cp[1].y - 3*cp[2].y + cp[3].y; float by = 3*cp[0].y - 6*cp[1].y + 3*cp[2].y; float cy = 3*cp[1].y - 3*cp[0].y; res.x = (int) ((ax * t3) + (bx * t2) + (cx * t) + cp[0].x); res.y = (int) ((ay * t3) + (by * t2) + (cy * t) + cp[0].y); return res; }
-- netd
The history section says that Pierre Bézier just used the curve, but [ here] I found that he actually discovered the curve independently: "This set of curves was discovered around the same time by two people: Bezier and de Casteljau. Bezier discovered it using the Berstein polynomials, while de Casteljau found a geometric representation". Is it right? Rodrigo Rocha 02:05, 16 Jun 2005 (UTC)
Am I missing something or should we adopt the convention that 00 = 1? Say we need to calculate the following Bernstein polynomial (for a given n, with i=0 and t=0):
Here we have 00.
00 = 0*0-1 = (1-1)*0-1 = 0-1-0-1 = 0
=> 0=1
(Which would be a valid argument if 0-1 were an element of *any* number field...) 94.0.147.195 ( talk) 04:11, 23 September 2010 (UTC)
I dropped "and there is no analytic formula to calculate the roots of polynomials of degree 5 and higher" for 2 reasons : Firstly, in most cases numerical approximations can be used instead of analytical solutions. But more importantly, it is seldom necessary to compute t. The most frequent operation is to compute the coeficients with Gaussian elimination if not directly. -- Nic Roets 09:18, 30 July 2005 (UTC)
I'd second dropping that phrase. There is no *algebraic* formula for the roots of a quintic, but there is a closed form solution in terms of hypergeometric functions (which are analytic). That is, there is no formula for the roots in terms of the coefficients using a finite number of additions, multiplications, and root takings (for quintic and higher order polynomials). There is, however, a formula involving a finite number of additions, multiplications, root takings, and evaluations of hypergeometric functions (for polynomials of degree greater than five, one needs generalizations of the hypergeometric functions). -- jimbo 00:44, 12 October 2005 (CDT)
Some curves that seem simple, like the circle, cannot be described by a Bézier curve or a piecewise Bézier curve
I have a different opinion. If you create a cubic Bezière curve which starts at [0,1],goes through [sqrt(1/2),sqrt(1/2)] and ends at [1,0] (its control points are [0,1],[x,1],[1,x],[1,0] where x=0.552285), the resulting curve is equivalent to a circle - the point distance from radius only differs in tenths of permille of the radius.--
janndvorakk 10:13, 1 January 2006 (UTC)
If there is some difference in radius, no matter how miniscule, then the curve cannot describe the circle. Dysprosia 10:24, 1 January 2006 (UTC)
I've added a section on constructing Bézier curves including some animated GIFs, because I think it helps to visualise Bézier curve formulae. I thought about adding a derivation so you can see from the construction how the Bézier formula comes about because someone was complaning "Wow, if only there was an explanation for the layman". For example something like this for a quadratic curve:
So doing some substitution...
I decided this was more appropriate to a high school textbook than an encyclopedia so I left it out. Twirlip 16:43, 27 August 2006 (UTC)
In main page I have change these formulae:
Correct formulae (Bézier curve is an interpolation between two degree Bézier curves):
The animations for the curves are too fast for anyone who actually wants to study them. Perhaps slow them down by about fifty percent or so.
Radical 05:34, 10 September 2006 (UTC)
These animations are great! Someone should nominate one as a featured animation. I never really understood bézier curves as fully as I should have before, but all the math is clear, without even reading the math. -- jacobus (t) 17:53, 12 October 2006 (UTC)
I agree, the animations are incredibly helpful.
70.22.140.232 22:09, 3 February 2007 (UTC)
As it turns out, the animations seem to indicate that the figures made with nails and string, as shown, for example, at http://www.stitchingcards.com/section.php?xSec=25 make the contour of a Bezier curve. (You'll probably think the next think I'll do is promote lava-lamps in articles related to convection or materials density change with temperature)
As for the layman understanding of what Beziers curves are or why are they so useful, I would like to suggest to change the order of the article so the explanations easier for the laymen go first. After all, few of us would have never learned there was something called a Bezier curve if it wasn't for their simple construction. -- DevaSatyam 12:07, 19 August 2007 (UTC)
I added a few links to my pages on how to construct Bezier curves geometrically. If you want to incorporate that into the main article, go right ahead. - SharkD 21:45, 30 October 2006 (UTC)
well.. i'm into high school, so i don't really have much understanding about bezier curves. But i was trying to plot a bezier curve in PowerToy Calc. I used the formula given in the talk section "Construction Bezier curves". This was what i used:
B(t) = ((1 - t)^2)*p + (2*t*(1 - t))*q + (t^2)*r
here, p, q and r were integral variables which i gave different values later. Now my doubt was.. whatever it is that i'm doing, is it right?? coz the curves which were plotted.. didn't really seem like bezier curves to me. If i'm not doing it right.. how is it done right?? i'd be grateful for the help.
Rohan2kool 10:47, 11 April 2007 (UTC)
In order to get a curve, p,q, and r need to be 2 dimensional vectors (i.e. points). If you just use scalar values, you'll probably end up with a parabola. I'm not familiar with PowerToy Calc, so I don't know what kind of support it has for parametric equations. Hope that helps.
mistercow ( talk) 01:11, 26 January 2008 (UTC)
The first reference Bézier curves, [1], redirects to [2], which is about Bézier surfaces. I wonder if perhaps this page [3] on Bézier curves was intended.-- Jwwalker 18:11, 10 August 2007 (UTC)
The C code we give for rendering cubic Béziers is very inefficient. Why don't we present the standard technique of repeatedly dividing the curve into two until each part is sufficiently close to a straight line? -- Doradus 14:55, 19 August 2007 (UTC)
Should say that one reason that cubic beziers are convenient in fonts (such as in PostScript Type 1 etc.) is that you can require that a cubic curve has two specified endpoints, and a specified tangent angle at each endpoint, and still have some further control over the shape of the curve between the endpoints. By contrast, if you use quadratic beziers, then specifying two endpoints and two corresponding tangent angles completely specifies the shape of the curve -- if you then want to change the shape of the curve, the only way is to recut it (using different endpoints), and this may have a further chain-reaction effect on neighboring quadratic curve segments (the dreaded non-locality problem when attempting to edit Truetype fonts in native format, i.e. without converting them). AnonMoos 18:47, 1 September 2007 (UTC)
Considering the Bezier curve was digitally popularized prior to Illustrator, etc. with PostScript, doesn't PostScript deserve a mention? Particularly as PostScript offers first-class citizen language entities for defining these curves. —Preceding unsigned comment added by 70.68.70.186 ( talk) 00:20, 16 August 2008 (UTC)
Yeah, just one more example of how PostScript just *might* be one of the most underrated and overlooked of all computer languages. Toddcs ( talk) 06:13, 31 August 2009 (UTC)
Hello, I corrected(niels@degooier.net) the Bezier Cubic formula from: B(t) = (1-t)3P0 + 3t(1-t)2P1 + 3t2(1-t)P2, t3P3, tE[0,1] to: B(t) = (1-t)3P0 + 3t(1-t)2P1 + 3(1-t)t2P2, t3P3, tE[0,1] Greetings from Holland :) —Preceding unsigned comment added by 212.29.183.137 ( talk) 12:30, 30 December 2008 (UTC)
Hi, is the headings for cubic and quadratic curves backwards? —Preceding unsigned comment added by 81.167.94.214 ( talk) 19:24, 7 May 2009 (UTC)
192.171.3.126 ( talk) 11:55, 12 May 2009 (UTC)In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms. Quadratus is Latin for square.
I don't think that this should be low importance- it is both heavily visited and important to Grasphics Editors like Inkscape and GIMP. Resident Mario ( talk) 18:55, 18 January 2009 (UTC)
For example:
To guarantee smoothness, the control point at which two curves meet must be on the line between the two control points on either side.
It would be nice to actually show "2 curves meeting" and the control point "at which they meet". Smoothness may possibly be intuited after illustrating the above.
I do see an illustration captioned "Example of two cubic Bézier curves patched together..", but it is not clear that there are 2 curves within. Am I just slow?
Jgsack ( talk) 20:07, 30 June 2009 (UTC)jim
The following text was removed: "In other words, the tangents in P0 and P2 both pass through P1". However that text is correct. The tangent lines to the Bezier curve at each of the end-points are lines through P1. Please explain why you think this is not true. − Woodstone ( talk) 00:05, 13 April 2010 (UTC)
Is a spiro spline a special bézier curve?-- 92.227.184.234 ( talk) 16:45, 21 July 2010 (UTC)
Using , and , the 2D bezier can be written in the form of:
and
or
and
where
and
We can find from the above equations:
By filling in into one of the two equations we find for the closed form:
or
I just removed the above section as it has a number of problems. It's a straightforward piece of algebraic manipulation, but even in the simplest case results in a far more complex expression for the curve that loses information from the x(t), y(t) form, such as the points given by t = 0, t = 1, etc., And is hardly worth doing as it's only one special case: 2D and quadratic. Therefore I suspect it's not from any source but is OR. It's unencyclopaedically written (too much first person mostly) and has at least two errors: trivial one in the first line and a more serious one that it doesn't handle a number of cases. There could be more as I can't check any source and can't be bothered to reproduce the steps for the last overly complex expression (which isn't even fully expanded which would make it even more complex). -- JohnBlackburne words deeds 14:30, 13 August 2010 (UTC)
I added a closed form solution for a quadratic bezier curve. However within minutes JohnBlackburne reverted my contribution as: Rv unsourced, non-notable result (OR?), unencyclopaedically written with trivial and non-trivial errors.
Let me comment to that.
Unsourced: It is unsourced as i derived it myself, seeing as i couldnt find the solution on the web. I can provide some more solution such that even you might be able to follow it.
Non-notable: Might be so, but nowhere on the internet I could find the solution or a clue to the solution. So at least I was looking for the answer. So there might be other ppl as well, so i thought id help other ppl out. Also you cannot both say its unsourced and non-notable. If its non-notable it shouldnt need a source.
unencyclopaedically written with trivial and non-trivial errors: Well unechyclopaedical maybe, maybe not. If you think you can improve my piece go ahead; trivial and nontrivial errors, please point em out to me or correct them.
On a side note, what is it with ppl that if they see some new addition that they want their mark as "I know what is best" and immediately remove it...Geez I try to contribute to the community and it gets removed within minutes. Next time i wont bother. Thank you very much. Good job —Preceding unsigned comment added by 88.159.78.227 ( talk) 14:34, 13 August 2010 (UTC)
To conclude, if you think its "own research" then fine leave it out. If you think its not-useful then thats your oppinion, somebody else might have another one. I think the closed form at least deserves a mention. —Preceding unsigned comment added by 88.159.78.227 ( talk) 15:35, 13 August 2010 (UTC)
I added the external link to my free Web e-book. It is a more complete treatment of piecewise interpolation for those new to the subject and aimed at graphics. I'll add references to other related Wiki pages in the future.
My e-book is both a study of the fundamentals described in simple terms as well as a reference showing many types of Polynomial Interpolation -both common types and some developed by the author. It also shows some techniques not seen elsewhere. Linear interpolation is looked at carefully and shown is as the basis for all more advanced types using only Algebra. Only after understanding how adding squared and cubed terms cause smooth curves, are the more advanced curves examined such as Bezier, Catmul-Rom, b-spline, and Hermite. More advanced mathematical concepts and notations are kept to a minimum. Some additional techniques that are suggested by the mathematics and that the author has not seen elsewhere are examined. A reference is also included with all the most common curve drawing methods and includes drawings to allow comparison with other types.
WHY: I had hoped, but failed to find a book explaining polynomial interpolation basics and thought that one must certainly exist with a collection of interpolation types. I found either purely mathematical tests or advanced graphics texts. Several years later, I started reading the original Internet Usenet "groups", comp.graphics.algorithms, and did lots of searching on the net. I saved whatever I found related to splines and curves, but did not look at it or try to understand any of it until early in 1996, I decided to look at what I had collected, and started to figure things out. I begin recording my thoughts for future reference and this is the result. The very book I wanted originally is now freely available for others.
I do not believe this has any issues with the Wiki self cite guidelines. -- Steve -- ( talk) 22:50, 28 August 2011 (UTC)
After reading through the comments I want to know if an example of 3D Bezier curve is possible. To my mind a helix is an example of 3D Bezier curve, my other observation suggests to me that all lines in the universe (space) are 3D Bezier curves and so can I say that there is no straight line possible in space? This is based on my understanding that a point in this Universe has always three dimensions. Comments please. Pathare Prabhu ( talk) 03:35, 3 April 2012 (UTC)
Could some mathematician add to this section an explanation of how a lower order curve changes as more points are added (i.e. quadratic curve through 6 points)? I went through many websites before I could finish my program, which is still ad-hoc. I suspect there is a very pretty explanation of how different bernstein polynomials are chosen in this case, and this math would help programmers from going astray. Thanks. — Preceding unsigned comment added by Greg Fichter ( talk • contribs) 15:36, 30 December 2013 (UTC)
A curve can be split at any point into two subcurves, or into arbitrarily many subcurves, each of which is also a Bézier curve.
I would like to modernize the terminology in the article in a few places. In particular, I would like to replace references to "straight line" with simply "line," since lines are always straight in modern terminology, and seeing the term "straight line" can give readers the confusing idea that lines may not always be straight. But I don't want to do this without giving the custodians a chance to comment ... Eleuther ( talk) 00:51, 13 September 2016 (UTC)
Even in the field of mathematics a line is simply a one-dimensional entity. Not necessarily straight.− Woodstone ( talk) 09:19, 19 September 2016 (UTC)
I have removed the second part of the sentence "The mathematical basis for Bézier curves—the Bernstein polynomial—has been known since 1912, but its applicability to graphics was not realized for another half century." -- this is a triviality of the same kind as in "Cryptography has been known since 1900 BCE, but its applicability to internet banking was not realised for another 38 centuries." -- Alexey Muranov ( talk) 16:21, 25 September 2017 (UTC)
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Hello! This is a note to let the editors of this article know that File:Bézier 2 big.gif will be appearing as picture of the day on 23 October 2018. You can view and edit the POTD blurb at Template:POTD/2018-10-23. If this article needs any attention or maintenance, it would be preferable if that could be done before its appearance on the Main Page. Thanks — Amakuru ( talk) 12:47, 16 October 2018 (UTC)
When originally viewing this page, I had hoped to see information related to the matrix representation of the spline. See: Bézier curvatures as matrix operations. -- SomeTimeLater ( talk) 17:40, 24 December 2021 (UTC)
Introduction needs to be clearer, for the general (non-computer scientist) reader. I think it is important to mention right at the start that the curve is found using a computer algorithm, with the control points as the input. Hence the previous version saying: "A set of discrete control points defines a smooth continuous curve, found by an algorithm."
Saying simply the curves are defined by control points is incomplete. How these control points define a smooth curve is left as a mystery, for non-computer scientists (computer scientists may assume it is done by a code).
Regarding "in any case, control points can simply be chosen to define a curve, rather than vice versa.' All continuous curves are not strictly Bezier (i.e. you can 't find finite number of control points for any given curve- though you can find very good fits). Maybe it should be mentioned that most curves can be approximated very accurately with Bezier curves. Pskuri ( talk) 00:52, 17 February 2022 (UTC)
Pskuri ( talk) 00:52, 17 February 2022 (UTC)
My latest version was:
" Discrete "control points" determine a formula (a vector function) that defines a smooth continuous curve. Bézier curves can approximate a wide variety of shapes including those found in the real-world that have complicated or unknown mathematical representations."
I think my version is a) More succinct. b) I am not sure they are "usually intended" to approximate real world shapes, though they ofc they can be used for that. c) All the equations that define the curve used in the article are vector functions that are determined by the control points (see: /info/en/?search=B%C3%A9zier_curve#General_definition)
Ignoring c), I thought a prior edit was just a more succinct version of yours:
"Discrete "control points" with a formula defines a smooth continuous curve. Bézier curves can approximate a wide variety of shapes including those found in the real-world that have complicated or unknown mathematical representations."
Thanks Pskuri ( talk) 21:36, 17 February 2022 (UTC)
An alternative more succinct version (accepting my point regarding c)
" Discrete "control points" determine a vector function that defines a smooth continuous curve. Bézier curves can approximate a wide variety of shapes including those found in the real-world that have complicated or unknown mathematical representations." Pskuri ( talk) 21:41, 17 February 2022 (UTC)
I don't think "usually intended" is appropriate since we don't know what drafters intentions are. Also in many cases like cars for instance they are not trying to approximate real world objects. They are creating new curves.
I spent a long time make those two sentences succinct and clear. Shall assume you weren't belittling my sincere efforts by calling it "verbiage"?
All the equations that define bezier curves are vector functions. Each bezier curve has its own vector function which is written in terms of the position vector of the control points.
I propose for now use my prior version:
"Discrete "control points" with a formula defines a smooth continuous curve. Bézier curves can approximate a wide variety of shapes including those found in the real-world that have complicated or unknown mathematical representations."
This uses fewer words than current version and also gets rid of the problem of asserting usual intent.
I suggest having another editor with mathematics background to review my proposed definition:
"Discrete "control points" determine a vector function that defines a smooth continuous curve."
The reason I prefer this is because I found "formula" too vague, also I believe this is a very simple summary of the equations on the page. Many thanks Pskuri ( talk) 23:29, 17 February 2022 (UTC)
The part about approximating "real world" shapes applies to Bézier splines (a chain of Bézier curves), not so much to individual Bézier curves. Therefore that sentence in the lead may be confusing. Not all Bézier curves are "smooth". A third order Bézier curve may have cusp. What is left may be stated as: "A formula applied to a discrete set of control points defines a continuous curve."−
Woodstone (
talk) 04:08, 18 February 2022 (UTC)
The parameter in the vector functions that define the curves (in the article) is 't'. The position vectors act in a similar way to coefficients in scaler equations. (The sentence would be ok I think if we said "the control points are parameters to a Bezier generating algorithm", i.e. parameters in a computer programming sense)
Taking into account above Woodstone comments I would propose, for time being:
"Discrete "control points" with a formula defines a continuous curve. Bézier curves can approximate a wide variety of curves including those found in the real-world, that have complicated or unknown mathematical representations."
Pskuri ( talk) 05:34, 18 February 2022 (UTC)
How about ""Control points" with a formula defines a continuous curve'. This phrasing has fewer words than "A formula is applied...". I don't think 'set of discrete' is needed, since the default is to assume the number of control points will be finite. Regarding the point about Bezier-splines (I assume you meant Bezier splines not B-splines), how about this for opening paragraph
So how about:
A Bézier curve ( /ˈbɛz.i.eɪ/ BEH-zee-ay) [1] is a parametric curve used in computer graphics and related fields. [2] "Control points" with a formula defines a continuous curve. Bézier curves can be combined to form a Bézier spline, which are used to approximate a wide variety of curves including those found in the real-world, that have complicated or unknown mathematical representations. Bézier curves can also be generalized to higher dimensions to form Bézier surfaces. [3] The Bézier triangle is a special case of the latter. Bézier curve is named after French engineer Pierre Bézier, who used it in the 1960s for designing curves for the bodywork of Renault cars. [3] Other uses include the design of computer fonts and animation. [3]
Pskuri ( talk) 16:40, 18 February 2022 (UTC)
References
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A Bézier curve is a parametric curve used in computer graphics and related fields. Related to the Bernstein polynomial, it is named after Pierre Bézier, who used it in the 1960s for designing curves for the bodywork of Renault cars. Other uses include the design of computer fonts and animation. Bézier curves can be combined to form a Bézier spline, or generalized to higher dimensions to form Bézier surfaces. This animation shows a linear Bézier curve, the simplest type, generating a straight line between the points P0 and P1. This is equivalent to linear interpolation. Animation credit: Phil Tregoning
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