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Subject: Re: area of surface of revolution of 3D curve
Date: Fri, 15 Apr 2011 19:06:05 +0000 (UTC)
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"Prakhar" wrote in message <io8rdg$5bb$1@fred.mathworks.com>...
> How do you calculate area of surface of revolution of 3D curve? Though using integration I am able to calculate the area, but I would like to know a simpler method such as Pappus theorem for 2D curve.
> 
> Is the Pappus theorem limited to 2D curve or is the generalisation of Pappus theorem for 3D curve available?
> 
> I would also like to know that is there a theorem which says that the line about which surface of revolution of a given curve has minimum area should pass through the centroid of the curve?
- - - - - - - - - -
  Given the axis of revolution, transform curve point coordinates to cylindrical coordinates using that axis as the z-axis.  Then the curve of z and r (regarding r as always positive,) and ignoring azimuth, would constitute a two-dimensional curve and the Pappus centroid theorem would apply to that.  However it means finding the 2D centroid and the total 2D arc-length along this 2D curve which would surely require some kind of integration.  This 2D arc-length is not the same as the original curve's 3D arc-length.  At least a one-dimensional integration seems inevitable.

Roger Stafford