Gary
GaryVasco
Posts: 3,352 
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Post by Gary on Oct 14, 2016 4:27:43 GMT
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Post by Admin on Oct 14, 2016 15:33:08 GMT
Gary
It seems to me that Needham is refering to subsection 5 of section 1 on page 273-275 and especially to the paragraph just under figure 5 on page 274 which begins "Gauss originally defined...". This is where the idea of the principal curvatures is introduced and in paragraphs one and two of page 295 it is referred to again in subsection 2 of section III on page 295.
As you probably noticed I have not submitted a solution to this suggested exercise yet. After reading it I think Needham wants us to argue that it follows from the symmetry of the figure in 19a on page 295 that
$\widetilde{r}=$radius of curvature of the generating tractrix and
$r=$the segment of the normal from the surface to the axis.
Vasco
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Oct 14, 2016 15:54:43 GMT
Gary It seems to me that Needham is refering to subsection 5 of section 1 on page 273-275 and especially to the paragraph just under figure 5 on page 274 which begins "Gauss originally defined...". This is where the idea of the principal curvatures is introduced and in paragraphs one and two of page 295 it is referred to again in subsection 2 of section III on page 295. As you probably noticed I have not submitted a solution to this suggested exercise yet. After reading it I think Needham wants us to argue that it follows from the symmetry of the figure in 19a on page 295 that $\widetilde{r}=$radius of curvature of the generating tractrix and $r=$the segment of the normal from the surface to the axis.Vasco Vasco, That is also the way that I understood the problem. I just don't see where it follows nicely from symmetry. Gary |
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Post by Admin on Oct 14, 2016 16:05:36 GMT
Gary
I think if you are looking at the pseudosphere then by symmetry one of the principal radii of curvature must be the radius of curvature of the generating tractrix because how could you argue that it should be to one side or the other?
Vasco
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Oct 14, 2016 21:32:52 GMT
Gary I think if you are looking at the pseudosphere then by symmetry one of the principal radii of curvature must be the radius of curvature of the generating tractrix because how could you argue that it should be to one side or the other? Vasco Vasco, What is the principle of symmetry that you are invoking? How is it different from deducing that the radius of curvature of the tractrix must be the radius of the circle in the plane of the tractrix? The exercise appears to require that we argue for the correct side. Gary |
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Post by Admin on Oct 14, 2016 23:07:54 GMT
Gary
I see what you mean. I'll give it some more thought and get back to you.
Vasco
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Post by Admin on Oct 15, 2016 6:59:28 GMT
Gary
Consider a point $p$ on the pseudosphere, and the plane $\Pi$ containing the normal vector n to the surface at $p$. If we rotate $\Pi$ about n then the curvature $\kappa$ of the curve in which $\Pi$ intersects the surface will vary and generally speaking will have a maximum and minimum value.
From the symmetry* of the pseudosphere it is clear that when the plane intersects it in the generating tractrix, that the curvature must be a maximum or a minimum - one of the two principal curvatures, and the other must therefore be when the plane is at right angles to the generating tractrix.
* If the plane is through the generating tractrix and we rotate it through a given angle, either clockwise or anticlockwise, the curvature at these two positions must be the same, and so the curvature when the plane intersects the tractrix must be either a maximum value or a minimum value.
Also notice that in posing this exercise Needham writes "As with any surface of revolution, it follows by symmetry..." This could be applied to a sphere which is a surface of revolution (of a circle). And in this case, by symmetry again, we can argue that the two principal curvatures are the same and both equal to $1/R$.
Vasco
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