Thread Subject:

area of surface of revolution of 3D curve

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?

"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?
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  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