# NURBS are wavy

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#### Norman

##### Well-Known Member
HBA Supporter
from the Nurflugel mailing list

Donald Stackhouse wrote: said:
Harry writes:

Don't forget the P51D mustang as a beautiful aircraft

No argument from me on that one.

The P-51's visual grace was not by accident. It was one of the first aircraft whose shape was lofted using conic sections.

For those of you who are mathematically "rusty", conic sections are the shapes you get when you slice through a cone at various angles. They include ellipses (which includes circles, since a circle is just an ellipse whose major axis and minor axis are exactly equal), parabolas, and hyperbolas.

The common denominator in all of these is that the mathematical equations that define their shapes are all "second order" equations; in other words, the highest exponent is two. For example, an equation of a parabola might be y = x2.

OK, so why is this important? Well, in order for a curve to contain any inflection points ("wiggles"), it must have an order of at least three.

Splines, b-splines, nurbs, and the other mathematical equations commonly used by modern CAD systems are all third order or more. They can have wiggles in them, even microscopic ones that are too small to consciously see, but big enough to register subconsciously. They are also big enough to cause ripples and peaks in the pressures and velocities of air flowing along them, things that could cause undesired behavior, such as flow separation. Ugly to the eye often also means ugly to the airflow.

Since a second order curve (such as a conic section) is mathematically incapable of containing any wiggles, not even microscopic ones, it is naturally mathematically smooth. That makes it naturally pleasing both to the eye, and to the airflow.

One of the big reasons I continue to use my "antique" (90's vintage) CAD software instead of "upgrading" to something newer is because my old system has an absolutely beautiful package for drawing with conic sections.
Norman Masters said:
Well, I finally had time to check out Don's distressing revaluation on my friend's autocad system. Acad doesn't do all the conic sections but polyarc is a string of tangent arc segments so that should do a fair curve. I opened a drawing of a 4412 with the LE rad and slope. The airfoil was drawn with a polyline and curve fit to make a polyarc. I decurved the polyarc to get the original polyline and set sap mode to <end>. Then started <spline> and snapped to the polyline vertices. Then I set snap mode to <nearest> and drew a three point circle inside the leading edge. Then I curve fit the polyline and zoomed in to inspect the differences. There is some deviation near the LE but not anything like the horror of the older B-splines. The NURBS curve makes the LErad very slightly smaller than the polyarc. The two circles are very nearly tangent at the leading edge and have the same slope. As I panned along the line the errors were noticeably smaller downstream. However the spline weaves back and forth across the polyarc just as Don said it would. I'd like to believe that the waviness was within normal building tolerance but one thing really stands out. The errors are biggest at the leading edge which is opposite of the requirements for laminar flow. The large number of points of modern ordinate sets probably fixes this but splines were originally developed so that we could get fair curves (including conic sections) in CAD with few points. NURBS are probably okay for computer generated airfoils with huge numbers of vertices but for antique sections I'm staying with good ol' polyarc, I know it's fair

--

.~.
/V\
// \\
/( )\ http://profiles.yahoo.com/libratiger62
(^^)-(^^) Norm

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#### orion

##### R.I.P.
This probably needs a bit of a simple historical side note. Prior to the incorporation of computers for lofting, all aspects of aircraft layout were done using pretty much the same methods and tools as used for ship-building. Even our terminology of "loft lines", "water lines", "butt lines" and "stations" comes from ships' loftsmen of centuries past. The tools used for the trade were identical to those used well into the sixties and even the seventies. Most of the shapes our industry used for generating the airplane loft were also based on the historical predecessors.

In the late thirties and early forties (apparently there is evidence that the methodology was used even earlier although I haven't seen it) the practice of incorporating conic sections became more common as the shapes delivered cleaner, more predictable shapes. But the mathematics also allowed for improved shape control and improved the generation of transitional surfaces and boundaries.

For a while there was a push to incorporate conical based lofting tools into CAD systems however I'm not sure how far the effort went since by the time the CAD tools became more prevalent in the industry, virtually all the computerized layout tools used some form of splines and blended curves - no mention was ever made of conics. Somewhere in my files I think I still have an old General Dynamics manual for conic lofting, but that's the only source I ever ran across that mentioned the subject and provided tools for its incorporation into the lofting (manual) process.

Now, the tools of the trade for the loftsman generally consisted of a series of "French" curves, which were used to connect a series of predefined points in order to arrive at the necessary shape. Up until the incorporation of the conic based techniques, it was pretty much the loftsman's eye that controlled the final outcome of the curves and/or surfaces. After that point the shape was more a mathematical function, but it was still the loftsman that made all this work.

This also applied to the lofting of the early airfoils. The geometry was generated using the ordinates provided by NACA or other sources, but the final shape was still the function of blending through the use of various curve templates. Given the mathematical tools of the day, the ordinates had limits to their accuracy - the numbers did not go out to too many decimal places. But then with French curves as the lofting tools, they didn't need to.

Taking the same ordinates an placing them into a computer today generally results in the wavy lines that are mentioned above. Actually, even if you use conics you're not going to get a perfect continual loft. The arcs may connect but the shape is only as good as the data.

In lofting with any form of splines, there is always a possibility of incorporating some form of waviness - this is why we often have curve and surface analysis tools that allow you to visualize if the loft is off. To minimize the chance of getting a weird curve or surface one must use fewer points, not more, and let the system's own mathematics drive the curve. If you try to place too many constraints on the shape, the program will work at meeting all the constraint points, which if they're not accurate enough, will drive the shape to be something that you don't want.

And this is true in airfoil sections also, especially if you're working at a shape whose database comes from point lists with insufficient accuracy. Yes, it takes a bit more work in developing the critical areas like the leading edge but once you get a good grip on the technique, the accuracy will be there.

#### RonL

##### Well-Known Member
Hi Norm
Looking at your yahoo link, it was thrilling to see the quote from "Nikola Tesla 1900".
I recently watched a PBS program about Da Vinci, and have an older PBS recording about Tesla, and several books about Tesla.
To me it seems that Da Vinci drew from all he saw, while Tesla was able to see in his mind, what can not be seen with the eyes, it makes me wonder why, after their works are made public, it still takes so long for their thinking to be accecpted?

It is my belief that in the Tesla Turbine, patented 1912 or 13, that we have the solution to much of our energy needs, (Thoughts of the, Vortex tube cooler, the Turbine, and Electric generator) the combination of these, in my thoughts, have the potential of extraction, and use, of large amounts of heat, that can be used for power, but are generally discarded as waste heat.

The recent threads about power plants, and links to different sites, that show what others are doing along power plant designs, to me makes the words of Tesla so much more powerful.

RonL

#### Norman

##### Well-Known Member
HBA Supporter
Nicola Tesla was one of the great minds of the 19th century. Unfortunately, like many of his contemporaries, he couldn't make the transition from the way the universe was understood pre & post Einstein. In spite of not understanding relativity or quanta his patents were instrumental in the development of multiphase electricity, AC electric motors, radio, etc. Everything that he invented that is still useful is being used. You and others are welcome to play with the stuff that has become obsolete if you want to. I'm content to look at the parts of a water pump and realize it's a direct descendent of that 1913 patent. Those guys who are trying to duplicate Tesla's performance claims for the boundary layer turbine have been saying "we're real close now" for years. I don't know if they're self deluded or just too emotionally (and financially) invested to admit that they've been chasing a phantom. Like any other prolific inventor some of his ideas worked better than others. The turbine is one that didn't turn out quite the way he'd hoped. I included that quote on my yahoo ID because I was reading "Master of Lightning" when I signed up and just thought that it was remarkable that he said that 69 years before the invention of the internet and nearly 100 years before it connected every country on earth

#### Dana

##### Super Moderator
Staff member
AutoCad is a very poor choice for this kind of work. It started life as a 2D program mainly for architechural drafting, and its tools for anything other than basic curves are limited at best. A CAD package with good surface modeling tools is best (I personally use KeyCreator and its ancestor Cadkey/FastSurf).

One point: Wavy CAD splines (or surfaces) can often be caused, counterintuitively, by having too many control points... small errors get magnified if the points are too close together; it forces the slope of the curve into something unreasonable and "flings" it out where you don't want it.

-Dana

No trees were harmed in the transmission of this message. However, a rather large number of electrons were temporarily inconvenienced.

#### Norman

##### Well-Known Member
HBA Supporter
Autodesk severed its 2D roots back in 1990 and NURBS were introduced around 96 or 97 but that's beside the point. No 3D or solid modeling were used. Just two different curves based on the same minimal set of control points. I am aware of the problem of too many points increasing the likelihood of errors, that's why I used an antique set of ordinates. Yes, I know they're not all that precise, but what I was after was the difference between the two curve types given the same set of ordinates not the absolute accuracy of the either. What I found was that the spline crosses the polyarc several times. The wave amplitude is proportional to the local radius with wave highest ranging from 0.0001 on the bottom surface where the radius is many times the chord to 0.048 on the leading edge radius. Most of the waves that I checked were ~.005, well within tolerance. It's that LE error that bothers me. Since I didn't scale it the comparison airfoil has a 100 unit chord. Assuming the units are inches .048 is over a millimeter. Even on an old fashioned, non laminar, airfoil that's more than enough to render the published performance data suspect. So which type of curve should we use to get the shape of that 1931 wind tunnel model? It's not just an academic question; many home builders are still looking at 4 digit sections.

So, is that .048 a dip or a bump? Funny, I drew in the circles for visual reference but didn't measure them so I could compare them to the published data. I'll check that tomorrow

You can do this comparison in any software you like. The results will be similar. The programers at Autodesk may be over paid but they're not incompetent

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#### Norman

##### Well-Known Member
HBA Supporter
AutoCad is a very poor choice for this kind of work.
The polyarc is within .005 of the published ordinates. The table only goes to 2 decimal places so I guess they rounded up. So that's a .048 dip:lick:

#### JerryFlyGuy

##### Well-Known Member
Interestingly enough. I've used conic lofting for a project I did a while ago. It's still waiting in the wings for a 1/4 scale model to fly but.. none the less..

I used the principles of conic lofting to do the fuse on this model [below] and it worked very well. The corner squarness of the fuse is controled by two control splines and the top and side profile are also controled by splines. This gave me excellent control when it came to curvature control and smoothness. It has 80 some algebraic equations imbedded into the model to assist w/ the conic solving when it calculates the profile from the control splines.

Alot has changed since I did this model in SolidWorks [the program used] so as future projects may not need this level of detail work to achieve a the same quality finish. SW has been working alot in these area's, I'm pumped to try some more hard core models in the near future!

I'm currently completing a CNC milling machine which will be used to machine the needed molds and plugs used to build a 1/4 scale test model. The milling machine has an 18 x 9 x 3 [all feet] working envelope and a repeatable resolution of less than 0.002".
I intend to build the first test model this winter and have it ready for flight testing next spring. At least thats the plan

Anyway, lofting has come a long way from the days of yore now w/ 3D cad tools its even easier. However, conic lofting is still a very useful tool and still a very powerful tool even when it's done w/ just a 2B and a pc of paper.

Jerry

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#### PTAirco

##### Well-Known Member
We will want to see pictures of that milling machine, when it gets done! I want one.

#### JerryFlyGuy

##### Well-Known Member
Sure, I would post one right now but I don't have my camera here.. I'll put one up later tomorrow night..

Jerry

#### JerryFlyGuy

##### Well-Known Member
Well here's a picture of the mill. It's not painted yet. It has a spoil board base which you can see painted white in the picture. The control is farther down the side of the machine. It's mounted on rails so it can traverse the length of the machine. You'll have to excuse the mess as well.. The shop needs a severe cleaning!

It has a 18K-rpm spindle [industrial not Dewalt] and is capible of 1000ipm rapids and cuts right up to the rapid speed depending on tool load [what it's cutting]

If you want one, I'll tell ya this much.. it's not for the faint of heart.. I've spent well over a years wages building it, and that doesn't include my time in the building or designing of it [I've got over 3000hrs into the build]. Some other spec's I was shooting for during the build. The entire length of the machine was leveled [at the rail mnt locations] to have less than 0.0005" deviation over any 12" of run. This means that total accumulated deviation over the 20ft length of the machine is less than 10thou..I'm convinced that it's less than this.. but there is no real economical way of measuring this. This may sound like alot [10thou] but in reality for the size of the machine it's relatively small. Taking into concideration that it's intended for machine molds and plugs for composite parts.. it's insignificant in the whole scheme of things.

It's been a fun, but tiring project. I'm not entirely sure I'd do it again, knowing what I do now..
I'm just glad it's this close to being done.. If I'm lucky it'll be painted this weekend..

Next project? A website for my company.. then a 5axis head for it. First I've gotta cast the raw parts for the head [and finish the design of it]. It'll be a year+ before I get the 5 axis up and running.. I'm not in a hurry anyway.. the CAM software to create 5 axis code is gonna cost me 1/2 what the machine is worth.. sigh.. machine tools are like airplanes.. to make a $million$, you need to start w/ two $million$..

Best
J

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