Talk:Non-uniform rational B-spline

Inline citations

I have added some inline citations to relevant sections from two of the main reference books ("The NURBS Book" and "An Introduction to NURBS with Historical Perspective") supporting the following aspects:

• Invariace of NURBS surfaces under affine transformations
• Construction of basis functions
• Definition of a NURBS curve
• Manipulation of NURBS curves

Nedaim (talk) 21:07, 5 March 2010 (UTC)

Image Caption

Excuse my ignorance - NURBS are new to me - but shouldn't the caption for the second image be "A three dimensional NURBS surface" rather than "A two dimensional NURBS surface"? The surface and its control points are defined in 3D space rather than 2D. If the 2D caption is correct, perhaps an explanation of the 2D/3D terminology [as it applies to NURBS] is justified. ScottColemanKC (talk) 13:53, 6 January 2009 (UTC) Good catch --SmilingRob (talk) 05:20, 28 January 2009 (UTC)

B-Splines vs. Bézier Splines

I'm not sure I like the identification "NURBS (Non Uniform Rational Basis, or Bézier Spline)" in the History section. A Bézier spline is a special case of a B-Spline; for example, a degree 3 Bézier spline on four control points is the same as a B-Spline with knot vector [0,0,0,0,1,1,1,1]. But I don't think it is standard to regard the B in "NURBS" as standing for "Bézier." See, for example, 3-D Computer Graphics: A Mathematical Introduction with OpenGL, by Samuel Buss, Chapters 7-8. I think it would be better to replace the above quote with simply "NURBS (Non Uniform Rational Basis Spline)". Verdanthue (talk) 23:33, 2 April 2008 (UTC)

I agree. Bezier splines have no internal knots, which rules out Non-Uniform-ness. Bringing them together in one

term makes no sense. I will follow your suggestion. Mauritsmaartendejong (talk) 20:55, 5 April 2008 (UTC)

Continuity again

I propose to remove the discussion on continuity within the article by splitting it in two distinct parts (and putting it in a separate section). The C0, C1, ... etc definitions relate to parametric continuity and require the derivatives of the curve with respect to the parameter to be continuous. G0, G1, ... definitions relate to geometric continuity, which I think is equivalent to parametric continuity provided that the parameter exactly represents the length of the curve -- which is hard to achieve in practice. In modeling, geometric continuity is usually a requirement. Parametric continuity is a bonus, e.g. if rendering can use parameter space directly for tesselation. Two separate NURBS curves which are not Cn continuous can be Gn continuous. For instance, two linear NURBS segments with different knot spans but collinear control points will show G1 continuity. Mauritsmaartendejong (talk) 15:11, 6 January 2008 (UTC)

Double knots in the circle example

"(In fact, the curve is infinitely differentiable everywhere, as it must be if it exactly represents a circle.)"

This is true in the geometric sense, is it true in the parametrized sense? Isn't it true that two rational functions which agree in all derivatives in some point must be identical, like it holds for polynomial functions? But here we definitely have pieces with different functions on successive intervals. Someone should do the calculations to check this.--130.133.8.114 (talk) 13:01, 30 June 2010 (UTC) G. Rote

Addition. I checked: the 2nd derivatives of the spline curve are not continuous. --130.133.8.114 (talk) 18:29, 30 June 2010 (UTC)

Since the segments are parabolas, with the same parabola for each segment, by symmetry it should be C¹ continuous at the double knots. Since it's quadratic, I'm not surprised the 2nd derivatives aren't continuous. —Ben FrantzDale (talk) 13:47, 16 September 2014 (UTC)

Clarification on the Number of Knots

"The number of knots is always equal to the number of control points plus curve degree minus one."

I urge the athor to expand his or her explanation of this relationship but I do not personally feel qualified to make these changes. I spent some considerable time on this issue and still am not clear on what the proper relationship should be. Using the Forenik tool, which is by the way very cool, referenced in the article I deduced that nKnts = nPts + nDeg + 1 whereas the article states that nKnts = nPts +nDeg -1. To me this implies that there is some confusion over whether the ends are or are not knots. In addition the number of knots issue needs to be clearly differentiated from the subject of end point multiplicity. To the casual reader multiplicity apparently reduces the number of knots.

And thanks much for this article. Between this article and the Forenik tool, I now have a very good understanding of NURBS. — Preceding unsigned comment added by Petyr33 (talk • contribs) 18:28, 21 February 2013 (UTC)

Number of Knots per Control Point

Quote from the text (from the section on Knot Vector): The number of knots is always equal to the number of control points plus curve degree plus one.

Shouldn't it be: ...always equal to the number of control points minus curve degree plus one..

Example: A cubic NURBS curve with 4 CPs has got 2 knots: 4 – 3 + 1 = 2 —Preceding unsigned comment added by 193.170.135.17 (talk) 11:05, 2 December 2007 (UTC)

No. If a cubic curve with four control points has two knots, they have multiplicity 4, bringing the total number of knots to 8 = 4 + 3 + 1. The multiplicity of the boundary knots causes the curve to be clamped to the first and last control point. See also property P3.2 in chapter three of Piegl and Tiller Mauritsmaartendejong 12:56, 2 December 2007 (UTC)

In my opinion the total number of knots (counted with multiplicity) should be equal to the number of control points plus degree minus 1. Exampled: degree=d=1 (order=2). These are just polygonal chains through the control points. clearly we don't need more knots than control points. And to get the curve starting at the first control point, we don't have to repeat the first knot more than d=1 times. (The first and last knot in the circle example should be deleted. Multiplicity 2 is enough (I did the calculation).)

It isn't a matter of opinion or of 'needing extra knots' to clamp the end points, it is a matter of how NURBS are defined! It is part of the mathematical definition of a NURBS curve that (including multiplicities) num(knots) = num(CPs) + order (CP is control point). Since order = degree + 1, the above CP+degree-1 formula runs counter to the definition of a NURBS curve. The correct definition ensures each CP has a 'fair chance' at the right number of knot spans. Consider that the number of knot spans affected by a CP is equal to the order (this is a consequence of the way the basis functions are defined). Consider a minimal cubic NURBS curve with 4 CPs: it is expected that the middle knot span is the only one affected by all 4 CPs. Also, since order=4, we know the first CP affects exactly the first 4 spans. So the last of those 4 spans must be the middle one! So, there is a middle span and three spans on either side, for a total of seven spans, or eight knots. This fits the +1 version of the definition. If we took off two spans on either end to fit the -1 version, then we'd lose the spans that are affected by only one CP! Note that all of this counts multiplicities as additional knots; a multiplicity just results in a 'degenerate' span, i.e., one with a zero-length interval in parameter space. --Migilik (talk) 05:20, 13 June 2013 (UTC)

Quote from 'Knot vector' "The number of knots is always equal to the number of control points plus curve degree plus one (i.e. number of control points plus curve order)", there comes 8 knots for a minimal cubic NURBS. This corresponds with OpenGL http://www.manpagez.com/man/3/gluNurbsCurve/ A little later in 'Comparison of Knots and Control Points': "there are groups of 2 x degree knots that correspond to groups of (degree+1) control points"So with the minimal cubic NURBS, 4CP correspond to 6 knots. By instance, if you want a bezier you may use knots = {0,0,0,1,1,1}. This corresponds with the "blossoming algorithm" for construction. I think this is a contradiction in the article. Also, I fail to understand why these two extra knots are needed, how do they affect the curve? — Preceding unsigned comment added by 190.31.220.128 (talk) 20:17, 28 May 2014 (UTC)

In general, the first d knots can also be distinct, but the curve will then not start at the first control point (it will start only (d-1)/2 "steps" later. --130.133.8.114 (talk) 18:51, 30 June 2010 (UTC) Günter Rote

Maybe the sentence should be refrased. All in all, I'd say it would be misleading to say that number of knots (NK) is ALWAYS equal to the number of control points (NCP) plus degree (DEG) minus one, because in most cases clamped curves have number of knots equal to the order at the ends. For cubic curves this means NK = NCP + DEG + 1 <=> 8 = 4 + 3 + 1 (Toivo83 (talk) 06:11, 2 November 2011 (UTC))

'misleading' is an understatement. NK = NCP + DEG + 1 is the correct form for all NURBS curves (regardless of whether the common 'open uniform' knot vector is used), and having NK = NCP + DEG - 1 in the article will likely cause serious confusion for people new to the field. --Migilik (talk) 05:20, 13 June 2013 (UTC)

Parameterisation ratio

Also note that the only significant factor is the ratio of the values to each other: the knot vectors [0 0 1 2 3], [0 0 2 4 6] and [1 1 2 3 4] produce the same curve.

Er, I've got a problem with the ratios (0:1 = 0:2 = 1:2) and (1:2 = 2:4 = 2:3) which is what this states. I can only conclude that the minimum value is also a significant factor. Can anyone help me out on this one?

The only significant thing is the ratio of neighboring parameter values in their ascending order, therefore the given example is correct. Minimum value has no significance. As long as a multiplier value exists to make ratio between different parameter values the same, the resulting b-spline curve will be the same.

E.g. 0 1 2 3 4 is the same as 0 2 4 6 8 (multiplier being 0.5)

or

0 3 6 9 12 is the same as 0 0.25 0.50 0.75 1.0 (multiplier being 12)

Actually for any uniformly increasing parameter values, curves with same control points position and same number of knot vectors will have the same shape (which means that all four above examples give identical curve shape provided their control points positions are the same).

For non-uniform splines:

0 0 1 3 4 5 is the same as 0 0 0.5 1.5 2.0 2.5 (multiplier is 2)

It is those differences in parameter values that determine chord length and affect the resulting curve shape. Therefore, same ratio between knot parameter values, combined with same control points positions will give the same curve. Think of it as a tension increased in one knot by any factor will not change the curve if the same factor of increased tension is applied to all the other knots.

The text doesn't make clear that changing the knot vector by a constant multiple will change the parametrization of the curve. The curve will be the same shape, but it will be different as a parametric function of u. Verdanthue (talk) 23:11, 2 April 2008 (UTC)

Uniformed circle?

Can someone explain what a "uniformed circle" is? --Doradus 03:21, 5 February 2006 (UTC)

I asked myself the same question. I think it's a typo, and should be 'uniform circle', which I think is a circle that will give equidistant points when the parameter space is uniformly traversed. As an aside, the table needs an order and a knot vector before it really defines a uniform circle. The order is three, and the knot vector something like {0,0,0,1,1,2,2,3,3,4,4,4} Mauritsmaartendejong 20:38, 19 June 2007 (UTC)

Picture

This article could do with a picture in as much as NURBS is an adjective to describe a class of shapes, particularly shapes of consumer products of recent years. —BenFrantzDale 16:46, 16 February 2006 (UTC)

---

Well, its kindoff hard and simple at the same time. See more is about the specifications of the image, i mean you could photograph any mobile telephone, or any western airplane. But more of the trouble here is to make a piucture that is MEANINGFULL for the potential viewer. See when i take a look at my telephone i know what im looking for so i can see how the patch layout is done. But the untrained person will most definetly not see it unless pointed out, and even then if the designer of the phone did a superb job there would be noothing to see. Just imaginary lines. So its definetly going to have to be a rendering.

Now to complicate this a bit, most eingeneering applications make use of as many independent pieces of trimmed nurbs surfaces that most noneingeneers wont believe. -J-

Subs versus nurbs

the following line needs editting:

Subdivision surface modeling is now preferred over NURBS modeling in major modelers because subdivision surfaces have lots of benefits.

Because it is simply untrue. Or it would be true if a modelling packge could be clearly demonstrated not to include cad packages, and/or if cad packages would be very specialized minority. But the simple truuth is that theres more cad users on nurbs out there than theres DCC visualisers with subdivision surface modellers. And yet no cad package has switched over to subds. Because theres NO 2 degree continuity... And the tools are severely lacking form engeneerig perspective. -J-

I do completely agree as this is more or less opinion. It implies that nobody uses NURBS any more and instead uses Subs. NURBS do have important advantages compared to Subs and therefor all mayor animation packages contain a NURBS engine. Opinions about Sub Division Surfaces beeing "better" belong to the appropriate page only. -R-

Miswording

When describing the alternative definitions for C0 through C2 (right after they are introducted) the following claim is made.

This definition is also valid for curves and surfaces with base functions higher than 3rd order (cubic). It requires that both the direction and the magnitude of the nth derivate of the curve/surface (d/du C(u)) are the same at a joint. The main difference to the definition above is the requirement for a same order of magnitude.

It sounds like whoever wrote this is confusing same 'order of magnitude' (roughly the same value) with having the same value. If not one of the definitions is stated incorrectly as nowhere do they require same order of magnitude. The order of magnitude bit appears in the next sentence as well.

I'm pretty confident that 'same magnitude' is what was intended from textual clues and my knowledge of the mathematical notions so I'm going to change it but I'm just learning about computer graphics so if I got it wrong please correct my error. In either case once a person who knows the facts comes along and reads this and the page over this comment will become superflous.

Logicnazi 09:17, 19 June 2006 (UTC)

Yes i changed magnitude in the first explanation to the word length, magnitude however does mean the same thing for a vector (magnitude of a vector is its exact length, so all vectors of same length have same magnitude). But for the readability its indeed better to use the word length. Because its a more understandable wording -J-

Edity, i think the entire section of C0 C1 and C2 continuity should be revised fully, indeed C0 continuity implies end points matching, C1 is matching the vector of change direction and C2 is matching vector length change. So it is NOT enough to be same length but the rate of change must also be a continuous function (remember we are talking of function continuation here)

So simply, c0 ensures that the curves continue seamlessly, c1 ensures the angles the same, c2 ensures change of rate is continuous. -J-

PS i was talking to a engineer who needed c3 continuity because of some stress calculational reason.

External links

Could someone, preferably with expertise in the subject, go through the last four (untitled) external links? Seems like a lot of duplicate information. --Ronz 00:27, 12 September 2006 (UTC)

Unverifiable History ("by linking to other websites only")

I removed the following sentence from the article and moved it here until it can be verified:

In 1993, the first interactive NURBS modeller for PCs, called NöRBS, was developed by CAS Berlin, a small startup company cooperating with the Technical University Berlin.

I did a Google search to try and verify it and found nothing of note. Dallben 22:33, 1 December 2006 (UTC)

87.123.118.104 01:39, 27 December 2006 (UTC)

As Google search is your only source for verification, you are much too daring to touch historic facts in this article. Wikipedia is not intented to be reduced to google-content only. You might have realized that 1993 was very long before Google was founded or active on the internet. If you doubt the facts here you should take a closer look at the history of this article, which includes additional information about the first NURBS modeler on the PC. And NöRBS was also listed on the 1995 published O'ReillyNew Riders Pub Book about 3D Studio Plugins. So please dont touch it again as you have nothing to offer which can be called a reason, but: >>I did not find it on a Google quicksearch.<<

87.123.118.104 01:39, 27 December 2006 (UTC)

• Well, I think it would be prudent for you to review Wikipedia's policies on verifiability (pay special attention to point three of the nutshell summary). Thus, you cannot claim that I am wrong in removing the sentence—it is unsourced and does little to add to the article's quality. So, instead of removing it, I've added a citation notice to the sentence, which you should fill out with nicely formatted sources, verifying the claim that NöRBS was indeed the first interactive NURBS modeller. (Please note that citing this article's history is patented nonsense—how on earth can that verify anything?). Finally, from your tone and from your IP address's geographic region, you've given me every reason to believe that a WP:COI is highly probable. Regardless of your motivation, cite the sources that verify this claim and the sentence will actually improve the article. Dallben 21:18, 27 December 2006 (UTC)

Hi, it seems we have a completely different reception of what is verifiability. Some editors have decided that the fact that NöRBS was published and presented first on the CeBit fare in 1994 is not an important information and therefor removed it from the article. You can find this information only in the history of the article now. I can perfectly understand people treating this information as not important enough to be part of the main article. In my reception this is a perfect source for verification: "It was developed in cooperation with the TU Berlin and presented on the biggest IT fare in the world, the CeBit 1994", as all these are verifiable facts. Not by a google search, but by researching at the place itself. Some editor added the fact that it was presented at the booth of a company called CAA Gmbh, to emphasize the credibility of the statement. This was removed too. Now you call it complete Bollocks to be directed to the history page for valuable information for verifiability?

Another - and in my view much more important - aspect is the fact that the article about the history and the use of NURBS containing the sentence you are challenging now is online since 10th3rd of July 2005. Many people have contributed and edited since then and much more have read and reviewed the content. There is not a single statement which could be regarded as doubt or even contradiction to the stated facts. Except yours, of course. Therefor the challenged informations have been steeled by the most valuable source of infomation: The Public. Everybody with access to the internet had the chance to say: Thats not true! for nearly two years now, and even more important: still has. Nobody did because it is simply the verifiable truth.

And we are not talking about some minor or even irrelevant subject, NURBS is one of the most widespread technologies in the world, even more than the PC as it is used in the industry too. Therefor it can be assumed that this article has been read by nearly anybody who has to do with NURBS on a professional level and knows about the Wikipedia.

All this is also the answer to the question: Was NöRBS really the first one on the PC? This is perfectly true until somebody comes up with some evidence saying: No, this one was earlier. Nobody knows everything in the world. I dont know if e.g. some Nepalese or Chinese climbed on the Mount Everest before Sir Edmund Hillary did so in 1953. Therefor, as long as there is no evidence for this case i will continue to say and believe: "Yes, it was Hillary who first climbed on top of this mountain".

But for those who still see any incompatibility of all the above with some *guidelines* here, the software called NöRBS was also presented (and included on CDROM) in the 1995 published book called >3D Studio Ipas Plug-In Reference<,

1. Author: Tim Forcade
2. Publisher: New Riders Pub; Bk&CD-Rom edition (July 1995)
3. Language: English
4. ISBN-10: 1562054317
5. ISBN-13: 978-1562054311

I am citing it here and not in the article itself for not making it subject of removement to the page history and in the following.... see above for details. 87.123.92.233 23:22, 3 January 2007 (UTC)

Removed the citation request as it is fullfilled here without adding unwanted overhead to the article itself. In addition the Google search cited here before shows that this page is used as a reference for other websites. Google itself judges this as a fact emphasizing the relevance of this page. 87.162.96.161 23:02, 7 February 2007 (UTC)

Removed the cleanup request too, as there has not been any noteable argument for keeping it since the request for verifiability has been answered here. 87.162.89.63 21:02, 11 April 2007 (UTC)

(Excerpt) "Vellum gives you the capability to create free-form curves through NURB (Nonuniform Rational B-) Splines, which provide more control over curvature than Bezier splines do, according to Ashlar." - Milburn, Ken (April 30, 1990). "Ashlar Vellum Gives Intuitive Interface to Macintosh CAD". InfoWorld 12 (18): 73–77. ISSN 0199-6649. — Preceding unsigned comment added by 204.128.192.31 (talk) 19:34, 7 July 2012 (UTC)

Request for image ?

From my point of view it does not make much sense to add an image or rendering, as this would mostly show the effect of rendering, which is not a subject of this article. Therefor it would not really help for understanding NURBS, but in opposition add to the widespread missunderstandings of NURBS by giving the impression of "this is how they look like". Instead look at any car built within the last decade, or any hairdryer, sophisticated shampoo bottle, designerphone or other industrially generated freeform object. There is a 99% chance you are looking at NURBS. 87.162.97.244 22:41, 19 April 2007 (UTC)

Violation

http://www.highend3d.com/dictionary/N/NURBS violation of rights, unauthorized copy please check dear friends After checking erase post. TNX Flaming-Balrog

This is definitely a (quite literal) copy. But, if I look at the substantial history of the wiki page, I see parts of the page gradually emerging, which suggests that the copy was made from wiki to there.

Mauritsmaartendejong 10:04, 24 June 2007 (UTC)

Application

I suggest to add this programm that is able to model NURBS XCModel. It was developed at University of Bologna, from Professor Casciola and his students under GNU/GPL licence. --151.51.25.112 (talk) 18:49, 3 January 2009 (UTC)

PCs in 1985 ? Impossible to do interactive modelling with these machines.

I heavily doubt the claim of Formation Design Systems of having offered "the first interactive NURBS modeller for PCs, called Macsurf (later Maxsurf)" in 1985 as PCs of that time did not offer the neccesary calculating power for interactive modelling with NURBS. The evaluation of the underlying mathematical equations is very complex and can NOT be done in an interactive way without the neccesary calculating power. My guess is they worked on mainframe-backuped workstations or terminals at that time and therefor now make a claim for "something looking like a PC". Or maybe they did some trials on the INTEL 80286 machines of that time which had a maximum speed of about 10 Mhz, or they regard "interactive editing of numbers" like you can do it in a texteditor as their idea of "interactivity". So this is a request to remove this claim unless they do not offer any proof for it.

basis functions

From bottom to top: Linear basis functions $N_{1,1}$ (blue) and $N_{2,1}$ (green), their weight functions $f$ and $g$ and the resulting quadratic basis function. The knots are 0, 1, 2 and 2.5

I "corrected" the caption according to my unexamined prejudice – you'll laugh, I know – that weight functions ought to add up to a constant, and (at least when used for interpolation) not drop suddenly from maximum to zero. A more knowledgeable person changed it back. Evidently I have much to learn. —Tamfang (talk) 01:58, 18 November 2010 (UTC)

Wrong Control Points in the Circle Example

The circle example has obviously been wrongly corrected by this person: (cur | prev) 06:35, 25 February 2011 Stefano.anzellotti (talk | contribs) (28,213 bytes) (→Example: a circle) (undo) Before, the control points were correct (namely only values from {-1,0,1}). The weights (in {1, sqrt(2)/2}) are still correct. I never edited something in Wikipedia... should I correct this or does some admin have to do this? — Preceding unsigned comment added by 137.226.115.43 (talk) 16:43, 1 September 2011 (UTC)

File:NURBS 3-D surface.gif to appear as POTD soon

Hello! This is a note to let the editors of this article know that File:NURBS 3-D surface.gif will be appearing as picture of the day on January 28, 2012. You can view and edit the POTD blurb at Template:POTD/2012-01-28. If this article needs any attention or maintenance, it would be preferable if that could be done before its appearance on the Main Page so Wikipedia doesn't look bad. :) Thanks! —howcheng {chat} 07:25, 26 January 2012 (UTC)

Picture of the day

Non-uniform rational B-spline is a mathematical model commonly used in computer graphics for generating and representing curves and surfaces for both analytic (described by mathematical formulas) and modeled shapes. Control points (shown as small spheres) influence the directions the surface takes. The square at the bottom sets the maximum width and length of the surface.Image: Greg L

Difference and similarities with bezier and bsplines?

I'm not an expert here, but I think a separate section detailing the differences and similarities with B-splines and Bezier curves would be beneficial. — Preceding unsigned comment added by 131.155.212.226 (talk) 12:23, 29 August 2012 (UTC)

I totally agree. The B-spline and NURBS pages aren't clearly differentiated. NURBS are an extension of B-splines, so everything to do with the basis functions and the knots really belongs on B-spline (unless I'm mistaken and regular B-splines don't include non-uniform knots). My understanding is that non-uniform knot vectors are a part of B-splines and so the difference between B-splines and NURBS is that NURBS are "rational" non-uniform B-splines. And to me, coming from computer graphics, "rational" means less to me than "homogeneous"; if I'd named them, I'd call them "homogeneous non-uniform B-splines". As far as I can tell, NURBS is just (non-uniform) B-splines with homogeneous coordinates, where you do a perspective divide at the end, as is typical in computer graphics. —Ben FrantzDale (talk) 13:58, 16 September 2014 (UTC)

First interactive NURBS history correction

"Real-time, interactive rendering of NURBS curves and surfaces was first made available on Silicon Graphics workstations in 1989. In 1993,"

There was a Vector-General GP-workstation with custom firmware developed to do real-time interactive NURBS for a presentation to the Ford Motor Co. prior to the above dates, about 1985-86. No orders were placed so the system did not go into comercial production. It was used to implement aplications and make videos of chasis-frame flexing and vibrations.

Lou schaefer (talk) 22:31, 23 December 2013 (UTC)

recursion formula knot vector

I must be pretty dense - the recursion formula ( Ni,n = fi,n Ni,n-1 .. ) blows up in the case of knots with multiplicity. Do you drop multiple knots as you recurse down in degree? Whatever the procedure, it needs to be spelled out in the article. — Preceding unsigned comment added by 2001:480:91:3303:0:0:0:6BC (talk) 21:49, 28 July 2014 (UTC)

Answering my own question: as an implementation detail when the denominators of f,g vanish due to multiplicative knots they can be set to any arbitrary finite value which ultimately drop out being later multiplied by zero. (assuming knots has permissible multiplicity) Someone with sufficient expertise should verify and add that to the article. The "{ ... , " notation ought to be explained as well. I guess it means the end knot might or might not be repeated.

Intuition of circle, conic section, and projection properties

Coming to NURBS, I was stumped for a long time about exactly why they have the useful properties they do. It seemed like magic. This article doesn't seem to address that issue directly. I'm tempted to Be Bold and add something to the intro, but I want to be sure my own intuition is correct first. I'm thinking of adding something like this:

NURBS have two particularly useful properties: NURBS can represent circles, circular arcs, and other conic sections; NURBS are also invariant under projection of their control points, meaning that given a NURBS, f, and a projection function, P, $P(f(t))=f'(t)$ where $f'$ is a new NURBS formed by transforming the control points of f by P. NURBS essentially solves both of these problems at once: The "rational" part of NURBs provides the projection properties: Essentially, NURBS control points are in homogeneous coordinates, where the "weight" is the homogeneous term and the normalization step of evaluating a NURBS includes a perspective division. The projection properties of NURBS lead directly to NURBS' ability to describe conic sections including circles and circular arcs: a quadratic B-spline has parabolic segments. Parabolas are conic sections. To produce other conic sections, we can simply project parabolas onto different planes.

Does that sound accurate? —Ben FrantzDale (talk) 13:40, 2 September 2014 (UTC)