![]() | This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||||||||||||||||||
|
Isn't this a special case of a b-spline?
In B-spline it says that Bézier spline are a special case of b-spline. So it could be made more explicit here. MathMartin 13:02, 29 Nov 2004 (UTC)
AFAIK Bézier splines are related to B-splines in that a Bézier can be represented as a B-spline and a B-spline can be represented as a sequence of one or more Béziers. However, this is more relevant to Bézier curves. Raichu2 7 Aug 2004
You could say that, but you could say that about many things. I don't think it's worth bringing it up in this article. As in, a bézier spline could be considered a special case of a NURBS too, you could also say a hermite spline is a special case of NURBS, and you can play this game with many many different splines, as lots of them have the capability to create the same shapes as others. What makes a spline unique isn't exactly the shape of the output itself, but rather how it transforms the control points, and what the continuity properties of the output piecewise funtion is like, as well as its shape. So I think it's almost a misnomer to ever call "splines" equivalent just because they can generate the same shape FreyaHolmer ( talk) 17:16, 30 July 2022 (UTC)
I've removed details that were relevant to Bézier curves, which are also known as Bézier splines. Raichu2 7 Aug 2004
The " Vector graphics" article redirects to this article under the name "Bezigon." However, the latter term is not defined in this article.
I am still not certain what a "bezigon" is supposed to be. I would guess that it is a geometric shape composed of multiple segments which are Bezier curves. Will an expert please add an appropriate section to this article? 66.168.74.119 23:07, 13 June 2007 (UTC)
I realize that this is only a stub but the article is way too gear-headed. Rigorous mathematical explanations are generally a good thing but, frankly, it is more important for math articles to include a brief segment that amounts to something like "In case you don't know much about this here's a short, practical description of this that you can use in everyday applications". In this case, providing a simple algebraic function that converts the real-valued control points and an interpolation variable into a real-valued interpolation result would make the article much more accessible (even that would be a subset of the general topic it would illustrate what is being discussed in a way that many more readers could understand it).
--- Mcorazao ( talk) 22:57, 16 May 2009 (UTC)
The article and subsequent discussions seems to lead to considerable confusion; especially to a reader who just wants to know what a "Bézier spline" is. It is my understanding that a Bézier spline is essentially a spline using Bézier curves, and not simply a synonym of "Bézier curve". The article as written might include that meaning, but would require interpretation. Perhaps a discussion of the distinction between a curve and a spline would be useful; especially in non-mathematical terms. 71.23.89.42 ( talk) 11:27, 12 July 2010 (UTC)
I suggest merging beziergon into Bézier spline. As far as I can tell, whenever the last endpoint on the last Bézier curve of a Bézier spline is placed exactly on top of the first endpoint of the first Bézier curve of that Bézier spline, the result is a beziergon, and all beziergons can be formed in that way. The difference between them is small enough that I think it makes more sense to have one merged article cover both kinds of splines, with a short paragraph mentioning the subtle difference. -- DavidCary ( talk) 20:45, 4 July 2011 (UTC)
The algorithm described in the General Case section is really poor. The results look horrible for large angles. It cannot create the approximation discussed above it in the Using Four Curves section. Even a cursory glance reveals that there is no way to generate the sqrt(2) value. The cited article for that section says it is for very small angles only. But, even for small angles, there are superior ways to do this.
It would be good if the
"Definition" would tell in a formal manner what a Bézier spline is, rather than presenting the silly claim that each spline were a sequence of
Bernstein polynomials, which would imply that every
polynomial, restricted to the closed
unit interval, were a Bernstein polynomial.—According to the introduction, a Bézier spline is a composition of Bézier curves. So it might be considered a
pair of
splines each of which has the same knots and is a "Bernstein spline", i.e., a spline whose polynomial components are Bernstein polynomials. The Bernstein coefficients in the respective splines form the control points for the single Bézier curves.—Splines usually are required to be
°highly differentiable" at the knots, therefore a Bézier spline may be required to satisfy a similar differentiability condition where composing Bézier curves meet. On the other hand, the introduction mentions the Béziergon that really has non-smooth corners ...
It would be good to clarify in the "Definition" whether Bézier splines and Béziergons are the same kind of thing or really very different things.—I do not want to edit the article because I don't know what terminology actually prevails. --
Uwe Lück (
talk) 09:55, 15 August 2013 (UTC)
All of the references in this article seem to be talking about a parametric x(t),y(t) shape -- -- where x(t) and y(t) are derived from a series the endpoints and control points stored in the font file (or some other source).
The parts of this article that lead people to think that a Bézier_spline has something to do with a curve or "a pair of splines each of which has the same knots" (?) are apparently misleading or incorrect.
How can we improve this article? -- DavidCary ( talk) 00:18, 5 December 2013 (UTC)
I've Done a rough cleaning. It needs more meat besides the application based on the aforementioned sources.
JMP EAX (
talk) 04:38, 19 August 2014 (UTC)
The sources I've look at diverge a bit on this. (See smoothness#Geometric_continuity for background.) I'm not entirely sure if the issue can be glossed over [as trivial reparametrization] or not. I'll have to read/think about it some more. JMP EAX ( talk) 17:37, 19 August 2014 (UTC)
The section "Using four curves" starts with "Considering only the 90-degree unit-circular arc in the first quadrant, we define the endpoints A and B with control points A' and B' respectively, as:
A (0,1)
A' (k, 1)
B' (1, k)
B (1,0)"
Since the unit circle is mentioned, I expected the arc to start on the x-axis at (1,0) and end on the y-axis at (0,1), thus traveling in counter-clockwise direction. It took me a while to realize that A (the starting point) and B (the end point) were defined oppositely. Thus A starts on the y-axis and the arc travels clockwise to B on the x-axis. I feel the reader should be told explicitly about the direction.
The general case section starts of with " Let the arc start at point A and end at point B placed at equal distances above and below the x-axis". Two questions:
These formulas are actually wrong. If anyone would check my work, I implemented the equations here: https://www.desmos.com/calculator/iidwqtcjzc I didn’t know where the author messed up, but I think it’s somewhere around equation 5.
While it is written that "The geometric condition for C2 continuity is C1 continuity, with the additional constraint that the control points are equidistant from the endpoint", it seems these conditions are not enough for C2 continuity of cubic Beziers. Given a curve $[a0, a1, a2, a3]$, the curve $[b0, b1, b2, b3]$ concatenates smoothly if:
$b0 = a3$ $b1 = 2*a3 - a2$ $b2 = 4*a3 -4*a2 + a1$ and you can choose $b3$
An editor has identified a potential problem with the redirect
Reticulating splines and has thus listed it
for discussion. This discussion will occur at
Wikipedia:Redirects for discussion/Log/2022 August 24#Reticulating splines until a consensus is reached, and readers of this page are welcome to contribute to the discussion. signed,
Rosguill
talk 15:28, 24 August 2022 (UTC)
![]() | This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||||||||||||||||||
|
Isn't this a special case of a b-spline?
In B-spline it says that Bézier spline are a special case of b-spline. So it could be made more explicit here. MathMartin 13:02, 29 Nov 2004 (UTC)
AFAIK Bézier splines are related to B-splines in that a Bézier can be represented as a B-spline and a B-spline can be represented as a sequence of one or more Béziers. However, this is more relevant to Bézier curves. Raichu2 7 Aug 2004
You could say that, but you could say that about many things. I don't think it's worth bringing it up in this article. As in, a bézier spline could be considered a special case of a NURBS too, you could also say a hermite spline is a special case of NURBS, and you can play this game with many many different splines, as lots of them have the capability to create the same shapes as others. What makes a spline unique isn't exactly the shape of the output itself, but rather how it transforms the control points, and what the continuity properties of the output piecewise funtion is like, as well as its shape. So I think it's almost a misnomer to ever call "splines" equivalent just because they can generate the same shape FreyaHolmer ( talk) 17:16, 30 July 2022 (UTC)
I've removed details that were relevant to Bézier curves, which are also known as Bézier splines. Raichu2 7 Aug 2004
The " Vector graphics" article redirects to this article under the name "Bezigon." However, the latter term is not defined in this article.
I am still not certain what a "bezigon" is supposed to be. I would guess that it is a geometric shape composed of multiple segments which are Bezier curves. Will an expert please add an appropriate section to this article? 66.168.74.119 23:07, 13 June 2007 (UTC)
I realize that this is only a stub but the article is way too gear-headed. Rigorous mathematical explanations are generally a good thing but, frankly, it is more important for math articles to include a brief segment that amounts to something like "In case you don't know much about this here's a short, practical description of this that you can use in everyday applications". In this case, providing a simple algebraic function that converts the real-valued control points and an interpolation variable into a real-valued interpolation result would make the article much more accessible (even that would be a subset of the general topic it would illustrate what is being discussed in a way that many more readers could understand it).
--- Mcorazao ( talk) 22:57, 16 May 2009 (UTC)
The article and subsequent discussions seems to lead to considerable confusion; especially to a reader who just wants to know what a "Bézier spline" is. It is my understanding that a Bézier spline is essentially a spline using Bézier curves, and not simply a synonym of "Bézier curve". The article as written might include that meaning, but would require interpretation. Perhaps a discussion of the distinction between a curve and a spline would be useful; especially in non-mathematical terms. 71.23.89.42 ( talk) 11:27, 12 July 2010 (UTC)
I suggest merging beziergon into Bézier spline. As far as I can tell, whenever the last endpoint on the last Bézier curve of a Bézier spline is placed exactly on top of the first endpoint of the first Bézier curve of that Bézier spline, the result is a beziergon, and all beziergons can be formed in that way. The difference between them is small enough that I think it makes more sense to have one merged article cover both kinds of splines, with a short paragraph mentioning the subtle difference. -- DavidCary ( talk) 20:45, 4 July 2011 (UTC)
The algorithm described in the General Case section is really poor. The results look horrible for large angles. It cannot create the approximation discussed above it in the Using Four Curves section. Even a cursory glance reveals that there is no way to generate the sqrt(2) value. The cited article for that section says it is for very small angles only. But, even for small angles, there are superior ways to do this.
It would be good if the
"Definition" would tell in a formal manner what a Bézier spline is, rather than presenting the silly claim that each spline were a sequence of
Bernstein polynomials, which would imply that every
polynomial, restricted to the closed
unit interval, were a Bernstein polynomial.—According to the introduction, a Bézier spline is a composition of Bézier curves. So it might be considered a
pair of
splines each of which has the same knots and is a "Bernstein spline", i.e., a spline whose polynomial components are Bernstein polynomials. The Bernstein coefficients in the respective splines form the control points for the single Bézier curves.—Splines usually are required to be
°highly differentiable" at the knots, therefore a Bézier spline may be required to satisfy a similar differentiability condition where composing Bézier curves meet. On the other hand, the introduction mentions the Béziergon that really has non-smooth corners ...
It would be good to clarify in the "Definition" whether Bézier splines and Béziergons are the same kind of thing or really very different things.—I do not want to edit the article because I don't know what terminology actually prevails. --
Uwe Lück (
talk) 09:55, 15 August 2013 (UTC)
All of the references in this article seem to be talking about a parametric x(t),y(t) shape -- -- where x(t) and y(t) are derived from a series the endpoints and control points stored in the font file (or some other source).
The parts of this article that lead people to think that a Bézier_spline has something to do with a curve or "a pair of splines each of which has the same knots" (?) are apparently misleading or incorrect.
How can we improve this article? -- DavidCary ( talk) 00:18, 5 December 2013 (UTC)
I've Done a rough cleaning. It needs more meat besides the application based on the aforementioned sources.
JMP EAX (
talk) 04:38, 19 August 2014 (UTC)
The sources I've look at diverge a bit on this. (See smoothness#Geometric_continuity for background.) I'm not entirely sure if the issue can be glossed over [as trivial reparametrization] or not. I'll have to read/think about it some more. JMP EAX ( talk) 17:37, 19 August 2014 (UTC)
The section "Using four curves" starts with "Considering only the 90-degree unit-circular arc in the first quadrant, we define the endpoints A and B with control points A' and B' respectively, as:
A (0,1)
A' (k, 1)
B' (1, k)
B (1,0)"
Since the unit circle is mentioned, I expected the arc to start on the x-axis at (1,0) and end on the y-axis at (0,1), thus traveling in counter-clockwise direction. It took me a while to realize that A (the starting point) and B (the end point) were defined oppositely. Thus A starts on the y-axis and the arc travels clockwise to B on the x-axis. I feel the reader should be told explicitly about the direction.
The general case section starts of with " Let the arc start at point A and end at point B placed at equal distances above and below the x-axis". Two questions:
These formulas are actually wrong. If anyone would check my work, I implemented the equations here: https://www.desmos.com/calculator/iidwqtcjzc I didn’t know where the author messed up, but I think it’s somewhere around equation 5.
While it is written that "The geometric condition for C2 continuity is C1 continuity, with the additional constraint that the control points are equidistant from the endpoint", it seems these conditions are not enough for C2 continuity of cubic Beziers. Given a curve $[a0, a1, a2, a3]$, the curve $[b0, b1, b2, b3]$ concatenates smoothly if:
$b0 = a3$ $b1 = 2*a3 - a2$ $b2 = 4*a3 -4*a2 + a1$ and you can choose $b3$
An editor has identified a potential problem with the redirect
Reticulating splines and has thus listed it
for discussion. This discussion will occur at
Wikipedia:Redirects for discussion/Log/2022 August 24#Reticulating splines until a consensus is reached, and readers of this page are welcome to contribute to the discussion. signed,
Rosguill
talk 15:28, 24 August 2022 (UTC)