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Post by Admin on Aug 16, 2016 11:40:49 GMT
Possible error, paperback, 2000 (with corrections), Chapter 6 exercise 3 line 3.
Gary
From reading exercise 3 and thinking about it what it is asking for, I think the statement of the exercise contains an error in line 3:
In the book, lines 1 to 3 contain the words:
"If $\widehat{C}$ is a great circle, then (18) says that $\mathcal{I}_C$ stereographically induces reflection of $\Sigma$ in $\widehat{C}$, but what transformation is induced if $C$ is an arbitrary circle?".
I think that they should read:
"If $\widehat{C}$ is a great circle, then (18) says that $\mathcal{I}_C$ stereographically induces reflection of $\Sigma$ in $\widehat{C}$, but what transformation is induced if $\widehat{C}$ is an arbitrary circle?".
Vasco
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Gary
GaryVasco
Posts: 3,352 
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Post by Gary on Aug 16, 2016 22:20:33 GMT
Vasco,
I commented in my original version of exercise 3 that there was a problem. I agree with your suggestion. I have to think about whether it is sufficient, that is, whether it is OK to think about a reflection of $\Sigma$ in $\widehat{C}$, or whether it should be a reflection of $\Sigma$ in $\mathbb{C}$. I am inclined to think your suggestion will be sufficient.
Gary
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Aug 17, 2016 3:16:40 GMT
Vasco,
I would like to suggest a change to the second to the last sentence of Ex. 3, so that it reads:
"$\widehat{w}$ is the second intersection point of $\Sigma$ with the line in space that passes through the vertex v and the point w" (rather than the point $\widehat{z}$).
Gary
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Post by Admin on Aug 17, 2016 5:33:06 GMT
Vasco, I would like to suggest a change to the second to the last sentence of Ex. 3, so that it reads: "$\widehat{w}$ is the second intersection point of $\Sigma$ with the line in space that passes through the vertex v and the point w" (rather than the point z). Gary Gary In my printing of the book it says "...the vertex $v$ and the point $\widehat{z}$.". Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Aug 17, 2016 17:01:43 GMT
Vasco,
Yes, but I am still suggesting that it should read w. I am thinking that perhaps he meant to write $\tilde{z}$. If it is not w, then there is some problem with my interpretation of the problem. Put another way, I don't think it possible for a line in space (as opposed to a line in $\Sigma$) to pass through v, $\widehat{w}$, and $\widehat{z}$. If "line in space" is to be read as "line in $\Sigma$, i.e. a great circle" then I might not suggest the change, though it is hard to imagine a line in $\Sigma$ intersecting with $\Sigma$, as Needham's phrasing seems to stipulate.
Gary
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Post by Admin on Aug 17, 2016 18:24:30 GMT
Vasco, I would like to suggest a change to the second to the last sentence of Ex. 3, so that it reads: "$\widehat{w}$ is the second intersection point of $\Sigma$ with the line in space that passes through the vertex v and the point w" (rather than the point z). Gary Gary In my printing of the book it says "...the vertex $v$ and the point $\widehat{z}$.". Vasco Gary OK but in your post quoted above you have put "(rather than the point $z$)", when it should be "(rather than the point $\widehat{z}$)" Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Aug 17, 2016 19:54:39 GMT
Vasco,
I fixed it.
Gary
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Post by Admin on Oct 2, 2016 15:41:16 GMT
Gary
Having published my solution to this exercise I think that the second last sentence of exercise 3 is correct as it reads in my copy of the paperback (year 2000), namely:
"That is, if $w=\mathcal{I}_C(z)$ then $\widehat{w}$ is the second intersection point of $\Sigma$ with the line in space that passes through the vertex $v$ and the point $\widehat{z}$."
Vasco
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