- Nov 28, 2003
- Grand Junction, Colorado
from the Nurflugel mailing list
Donald Stackhouse wrote: said:Harry writes:
Don't forget the P51D mustang as a beautiful aircraftNo 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
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