Surface Area Calculations in AutoCAD

[jonthebaptist]

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 piece-wise differentiable surfaces and said part is piece-wise differentiable. I have had my calculations double-checked 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.

 

[jonthebaptist]

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.

 

[xxChrisxx]

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.

Pretty much everything computer based uses discreet methods to calculate stuff. All 3D CAD packages I've used do.
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.

 

[jonthebaptist]

Pretty much everything computer based uses discreet methods to calculate stuff. All 3D CAD packages I've used do.
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.

I'll try to be clear without sounding presumptive. I used theorems from Differential Geometry that apply to piece-wise differentiable surfaces, and since the part is piece-wise differentiable, the theorems apply with mathematical uncertainty, meaning uncertainty set exactly to 0. The only places for error would be transcription, algebraic, or in one's conceptual understanding of Differential Geometry. Differential Geometry is much more powerful than the Analytic Geometry most engineers are used to. I know that Differential Geometry is not a class Engineers are required to take, but it may be worth it if your interest is in drafting or design.

Also, 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, arc-length, 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