
#1
Jul911, 11:31 PM

P: 17

My problem: I calculate, using Differential Geometry, the surface area of a specific part to be 50% more than the surface area AutoCAD calculates it to be using the AREA command on an extruded solid.
I am certain that my calculations are correct. I use theorems of Differential Geometry that apply to piecewise differentiable surfaces and said part is piecewise differentiable. I have had my calculations doublechecked by a colleague and verified by an independent calculation, so I am certain there are no typographical or algebraic errors. I used wolfram alpha to run the arithmetic in obtaining the final value. My Question: I would really like to be able to account for the deviation between my calculation and AutoCAD's. If anyone has any general information about the algorithm used in AutoCAD's AREA command, enough information that a Physics major who (barely) passed the undergraduate and graduate Differential Geometry courses could determine if such an algorithm applies to said part, I would be very grateful. Note: I have produced a (sloppy) proof that their exists no equiareal mapping from the surface of said part to any of the common primitives. 



#2
Jul1011, 08:22 PM

P: 17

I reevaluated my calculation and found an error. The new calculation matches AutoCAD's value. However there still is some interesting things to consider. The part is a helix, and a colleague originally calculated the surface area for this part by approximating the helix as a sum of circles, and his value also agrees with mine and AutoCAD's values. However, that approximation only applies because the pitch for this helix is small. So this still leaves the possibility that the AutoCAD was calculation was an approximation. So if anyone knows if AutoCAD calculates AREA using approximations with primitives known from analytic geometry, or if it uses the full power of Differential Geometry to calculate surface area, it may be useful knowledge for anyone else who is designing parts with complex geometries.
Note: It appears from my initial readings that constructive solid modelers such as Solidworks or Pro/E store the information as boolean sums of a library of known primitives, thus it is possible that they may not give accurate calculations for some surfaces. 



#3
Jul1111, 05:15 AM

P: 2,032

They use such small slices (on very high accuracy setting) that any error is negligable for most uses. It's certainly more accurate (for a given calcualtion time) than using assumptions and calculating by hand. 



#4
Jul1111, 06:33 PM

P: 17

Surface Area Calculations in AutoCADAlso, I'll note that the value from AutoCAD differs in the 3rd decimal place, well within our tolerance for this specific helix, but that does indicate that AutoCAD did use an approximation. Since the dependences of various values of interest, arclength, surface area, etc... on the parameters of the helix are not linear, there is no way to predict the deviation of any approximation (unless more specific knowledge about the approximation is given) if any parameter is changed significantly. So use caution if you are doing anything with helices, or other geometries that do not equal any Boolean sum of geometrical primitives known from Analytic Geometry. For some more basic information about the helix, see http://mathworld.wolfram.com/Helix.html 


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