Gary
GaryVasco
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Post by Gary on Aug 14, 2016 0:01:52 GMT
A warning to any readers: I am probably leaving a trail of wrong answers behind me in this chapter. In Exercise 3 in particular I find it hard to follow the reflections conjured up. The attachment is an attempt to represent the problem, but I can't resolve apparent contradictions in the question sufficiently to attempt an answer. This is the sort of problem where one would like to see the question put up on the board where one could ask for clarification.
[attached answer withdrawn for rewrite according to Vasco's suggestion in errata]
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Gary
GaryVasco
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Post by Gary on Aug 17, 2016 3:26:35 GMT
Vasco, I have rewritten Ex. 3 taking your suggestion in the Errata into account and adding a suggested change of my own. I look forward to seeing your take on this. nh.ch6.ex3.pdf (521.81 KB) Gary |
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Post by Admin on Oct 2, 2016 16:25:33 GMT
Vasco, I have rewritten Ex. 3 taking your suggestion in the Errata into account and adding a suggested change of my own. I look forward to seeing your take on this. View AttachmentGary Gary I have now published my solution. I have a different take on the exercise. Vasco |
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Gary
GaryVasco
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Post by Gary on Oct 4, 2016 3:42:59 GMT
Vasco,
I read your answer and commented on it somewhere. I still remain puzzled about the projection from a v off the sphere. I have rewritten my answer in a way that doesn't involve that odd vertex and its projection, but I think does satisfy the instruction to generalize (18) to reflections in arbitrary circles on $\Sigma$. When the ideas are plotted, they appear to confirm the analysis.
Gary
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Gary
GaryVasco
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Post by Gary on Oct 7, 2016 4:38:36 GMT
Vasco,
Two new perspectives have been added to the plot in Ch 6, Ex. 3 and small changes have been made following your suggestions.
Gary
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Post by Admin on Oct 7, 2016 6:33:16 GMT
Gary
I have just read your document. Here are a few comments:
Really like the new diagrams. In the second diagram you could draw the four lines from $v$ which are tangents to $\widehat{J}$ and $\widehat{G}$. I drew them with ruler and pencil on my printed copy and you can see immediately that they are tangential to the circles at their intersection points with $\widehat{C}$.
Point b): I think you need to say "...tangents to $\Sigma$ and orthogonal to $\widehat{C}$
Point e), line 2:.."they are orthogonal...". This sounds as though you are saying that $\widehat{G}$ and $\widehat{J}$ are orthogonal (to each other). They may be under some circumstances but we are trying to say here that they are both orthogonal to $\widehat{C}$.
On line 3 I think it is more accurate to say "...touches $\Sigma$ at the point..."
Point g), line 1: again you seem to be saying that $\widehat{G}$ and $\widehat{J}$ are orthogonal to each other rather than to $\widehat{C}$.
Your point of discussion at the top of page 5: Why do you say that because $\widehat{w}$ and $\widehat{z}$ are projections of $w$ and $z$ that this implies that $\widehat{w}$ and $\widehat{z}$ are inversions in $\widehat{C}$ on $\Sigma$? How does it follow? What reasoning are you using to arrive at this conclusion?
Vasco
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Oct 7, 2016 16:20:05 GMT
Gary I have just read your document. Here are a few comments: Really like the new diagrams. In the second diagram you could draw the four lines from $v$ which are tangents to $\widehat{J}$ and $\widehat{G}$. I drew them with ruler and pencil on my printed copy and you can see immediately that they are tangential to the circles at their intersection points with $\widehat{C}$. Point b): I think you need to say "...tangents to $\Sigma$ and orthogonal to $\widehat{C}$ Point e), line 2:.."they are orthogonal...". This sounds as though you are saying that $\widehat{G}$ and $\widehat{J}$ are orthogonal (to each other). They may be under some circumstances but we are trying to say here that they are both orthogonal to $\widehat{C}$. On line 3 I think it is more accurate to say "...touches $\Sigma$ at the point..." Point g), line 1: again you seem to be saying that $\widehat{G}$ and $\widehat{J}$ are orthogonal to each other rather than to $\widehat{C}$. Your point of discussion at the top of page 5: Why do you say that because $\widehat{w}$ and $\widehat{z}$ are projections of $w$ and $z$ that this implies that $\widehat{w}$ and $\widehat{z}$ are inversions in $\widehat{C}$ on $\Sigma$? How does it follow? What reasoning are you using to arrive at this conclusion? Vasco Vasco, Thanks for the suggestions. I have fixed them all. Regarding my reasoning about the inversion of $\widehat{z}$ and $\widehat{w}$ in $\widehat{C}$: When $z$, $w$, $C$ and the line, say $L$, though $z$ and $w$ are projected to $\Sigma$, $\widehat{L}$ is a segment of a great circle, so $\widehat{z}$ and $\widehat{w}$ appear to be reflected in $\widehat{C}$ and connected by $\widehat{L}$ in $S$. Stereographic projection from $\mathbb{C}$ to $\Sigma$ is one to one and conformal, so if they are inversions in $C$, they must be inversions in $\widehat{C}$. Gary |
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