Re:DOS Subsystem for Linux
by thaumasiotes on Saturday September 18, 2021 @04:36AM
(#28574631)
Hmm. Thinking about it as an integral of cross sections, the torus is just a cylinder with height equal to the circumference of the circle that defines the center of its tube. And I see that that is correct, at least for non-weird tori.
> [For a non-weird torus,] The volume of this solid torus and the surface area of its torus are easily computed using Pappus's centroid theorem
> These formulas are the same as for a cylinder of length 2πR and radius r, obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out (rectifying) the line running around the center of the tube.
> The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side.
Emphasis mine; that third sentence is weirding me out. Still thinking about it as an integral of cross sections, I can't see why the surface area or volume of the inner half of the torus should be different from the surface area / volume of the outer half. Are they?
https://en.wikipedia.org/wiki/Torus goes on to say this:
> [For a non-weird torus,] The volume of this solid torus and the surface area of its torus are easily computed using Pappus's centroid theorem
> These formulas are the same as for a cylinder of length 2πR and radius r, obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out (rectifying) the line running around the center of the tube.
> The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side.
Emphasis mine; that third sentence is weirding me out. Still thinking about it as an integral of cross sections, I can't see why the surface area or volume of the inner half of the torus should be different from the surface area / volume of the outer half. Are they?