Talk:Bézier curve

message: 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?
:Good question... I have no idea. Κσυπ Cyp   21:59, 15 Sep 2004 (UTC)
replies: text: It means you can't derive a second curve that is parallel to the first by mathematical operations on the control points of the first. It works for some cases, but not in general. Maybe the phrasing could be better, but it seemed clear enough to me.Graham 22:57, 15 Sep 2004 (UTC)
replies: text: What does parallel mean, when it's about curves? Is it a relation between two Bézier curves? Κσυπ Cyp   23:09, 15 Sep 2004 (UTC)
replies: text: Well strictly speaking parallel refers to straight lines or planes, so if we are speaking strictly, then 'parallel curves' is an oxymoron. However, in everyday language it should be clear what this means - a second curve that maintains a constant distance from the first at every similar point, like a pair of railway tracks rounding a bend. If there is a "proper" word for this I don't know what it is. The ability to do this is actually rather important in the real world, where bezier curves are not just mathematical curiosities but are a practical way of implementing computer graphics, etc. Creating a curve 'parallel' to another is often needed, but coming up with an algorithm for it requires some cunning, because of the mentioned limitation of the curve's essential properties. I made a stab at rewording the sentence, but 'parallel' is still in there ;-) Graham 23:47, 15 Sep 2004 (UTC)
thread_title: G1 continuity

message: 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)
thread_title: Merging Bezier curve & Bernstein polynomial

message: 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)
replies: text: Again this may be a layman vs. mathematician situation. Mathematically the bezier curve may be one dimensional, but intuitively it's two dimensional because it produces a point (x,y). I agree that the maths should be rigorous but I also think we musn't lose sight of the audience - which in general are not other mathematicians (who will have better references to work from than this), or even computer graphics programmers, but just people who are interested and looking for a simple introduction to the topic, but with enough info to go away and do something useful with it if they want to. That's my opinion anyway. Graham 23:52, 21 Sep 2004 (UTC)
thread_title: Dimension

message: 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):
:b_{i,n}(t):= {n \choose i} t^i (1-t)^{n-i} \qquad
:b_{0,n}(0):= {n \choose 0} 0^0 (1-0)^{n-0} \qquad
Here we have 00.
:Of course 00 = 1 !!
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)
thread_title:  0<sup>0</sup> 

message: 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)
thread_title:  Dropped roots comment 

message: Some curves that seem simple, like the circle, cannot be described by a Bézier curve or a piecewise Bézier curveI 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)
replies: text: The Bézier spline article and its corresponding talk page is a better place for describing a Bézier spline that looks like a circle. --DavidCary (talk) 22:03, 8 August 2011 (UTC)
thread_title:  Circle description 

message: 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:
\mathbf{Q}_0(t) = (1 - t)\mathbf{P}_0 +
t\mathbf{P}_1 \mbox{ , } t \in [0,1]
\mathbf{Q}_1(t) = (1 - t)\mathbf{P}_1 +
t\mathbf{P}_2 \mbox{ , } t \in [0,1]
\mathbf{B}(t) = (1 - t)\mathbf{Q}_0 +
t\mathbf{Q}_1 \mbox{ , } t \in [0,1]
So doing some substitution...
\mathbf{B}(t) = (1 - t)((1 - t)\mathbf{P}_0 + t\mathbf{P}_1) +  t((1 - t)\mathbf{P}_1 + t\mathbf{P}_2) \mbox{ , } t \in [0,1]
\mathbf{B}(t) =
(1 - t)^{2}\mathbf{P}_0 +
t(1 - t)\mathbf{P}_1 +
t(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]
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)
replies: text: Wow, your section rocks. The article was understandable enough, but your section really finished the deal (especially the sexy animations.) Thank you JakeParker 02:07, 2 September 2006 (UTC)
thread_title:  Constructing Bézier curves 

message: 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)
replies: text: How's that? Twirlip 20:15, 12 September 2006 (UTC)

text: I also agree, and have suggested the animations for WP:FPC at WP:PPR (Wikipedia:Picture peer review/Bézier curve animations). --Lph 01:09, 11 February 2007 (UTC)

text: This animations finally helped me understand what is so special about Bezier curves.  The small size of the sample C code also gives a hint to its importance, they are much easier than the formula that appear so prominent at the beginning of the article.  I would even add that I was lucky to skip over those formula and got to the animation and code section before dismissing the article and remain in the dark.  
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)
thread_title: Animations

message: 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)
thread_title: New links

message: 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)
thread_title:  a bit of doubt... would be thankfull if answered 

message: The first reference Bézier curves, , redirects to , which is about Bézier surfaces.  I wonder if perhaps this page  on Bézier curves was intended.--Jwwalker 18:11, 10 August 2007 (UTC)
thread_title:  Reference link in doubt 

message: 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)
replies: text: I have removed it.  De Casteljau's algorithm is much more efficient.  --Doradus (talk) 01:22, 14 December 2007 (UTC)
thread_title:  The code for rendering 

message: 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)
thread_title: Cubic beziers in fonts

message: 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)
replies: text: Does anyone know why Bezier curves were chosen for Postscript originally? (Would that be "DAK" in text-speak?)  Are they mathematically easier to evaluate (less computation intensity) than other polynomial curves, or allow a better "detail/graphic-ability/computation-load" ratio?  Or were they already used in specialized applications when Adobe created the Postscript standard?  A citeable source would be nice, but would probably be hard to come by.  Maybe something on folklore.org.  (Probably not citeable, but as an external link maybe)  Thanks. Jimw338 (talk) 18:17, 29 September 2019 (UTC)
thread_title: PostScript?

message: 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)
thread_title: Cubic Bezier, corrected formula

message: 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)
replies: text: You missed a heading, so I added one. No, the names are not backwards, although it does sound like they are. The equations are polynomials, and the names come from their degree - linear, quadratic, cubic, quartic etc. I can see how "quadratic" sounds like it should mean "of degree 4", but it doesn't, it means "of degree 2". To quote the Wikipedia disambiguation page: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.192.171.3.126 (talk) 11:55, 12 May 2009 (UTC)
thread_title: Names of the Curves

message: 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)
replies: text: It's the math project that has rated it low, meaning it has a low priority with respect to the goals of the project. Other projects are free give it a different priority.--RDBury (talk) 22:25, 6 June 2010 (UTC)
thread_title: Low Importance?

message: 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)
thread_title:  Please illustrate concepts a bit more. 

message: 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)
replies: text: Shouldn't the dots be removed, as this is regular products, not dot-products? I'm reluctant to do it, since this i s not my area of expertise, but it baffled me. Have I overlooked something? 72.48.211.208 (talk) 04:31, 20 February 2014 (UTC)
replies: text: Dots can also be used for scalar multiplication, but scalars should not have been boldface. −Woodstone (talk) 16:19, 20 February 2014 (UTC)
thread_title:  Derivative 

message: Is a spiro spline a special bézier curve?--92.227.184.234 (talk) 16:45, 21 July 2010 (UTC)
thread_title:  Spiro Spline 

message: 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). --JohnBlackburnewordsdeeds 14:30, 13 August 2010 (UTC)
replies: text: Well, it's useful for showing that a quadratic Bezier curve is a segment of a parabola, since the discriminant of the conic section is 0. And since any 3 points lie on some plane, even in higher dimensions, we can find a plane that the 3 points as well as the quadratic Bezier curve lie on, so the properties still hold. --24.130.148.132 (talk) 21:07, 27 April 2013 (UTC)
thread_title:  Removed section 

message: * I've logged on just to stay that Twirlip's animations are simply awesome.  Kudos for your contributions, Twirlip.  Great stuff indeed. -- Mecanismo | Talk
:* You're welcome :) Twirlip (talk)  —Preceding undated comment added 19:05, 24 February 2011 (UTC)
thread_title:  Awesome animations 

message: 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)
thread_title:  External Links 

message: 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)
replies: text: Yes 3D Bézier curves are possible. Their definition is identical to the ones given, but the vectors Pi should then be points in 3D space. A circle, or part thereof cannot be written as a Bézier curve, so neither can be a helix. On the contrary, a straight line segment can be represented by a Bézier curve of any order, by having all control points on the line. −Woodstone (talk) 08:51, 3 April 2012 (UTC)
thread_title: Bezier curve in Universe

message: 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)
replies: text: Why are you so obsessed with removing this harmless and informative remark, on such senseless grounds? You seem to object to it based on a willful and obtuse misreading of what it actually says. In particular, you seem to think that, because there were no computers in 1912, there was no graphics in 1912. But people had been creating graphics based on mathematical formulas for centuries before 1912. The remark simply says that Bernstein polynomials weren't considered in this role until ~50 years after they were first defined.  Eleuther (talk) 14:23, 28 September 2017 (UTC)
replies: text: Mentioning a purpose a specific tool was not used for at some time, is an open door to infinite expansion. Save it till the point where it is used. −Woodstone (talk) 12:19, 29 September 2017 (UTC)
replies: text: Thanks, Woodstone, I will re-apply the remark in a form that I hope will satisfy everyone involved. Eleuther (talk) 13:23, 30 September 2017 (UTC)

text: I didn't find any application of Bézier curves to any graphics, except computer graphics.  (If i understood correctly, their use by Pierre Bézier was computer graphics.)  I will appreciate any references to any examples where someone drawn some Bézier curves by hand, without using computers.  I mean in the last half century, when their application to "graphics" is well known. --Alexey Muranov (talk) 21:29, 15 May 2018 (UTC)
thread_title:  "Applicability to graphics" in 1912-1962 

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thread_title:  External links modified 

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message: 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)
thread_title: Matrix Representation

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