Gary
GaryVasco
Posts: 3,352 
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Post by Gary on Feb 14, 2017 8:36:18 GMT
Vasco, I think there is a mistake in Exercise 25. It is explained in my answer. nh.ch6.ex25.pdf (86.63 KB) Gary |
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Post by Admin on Feb 14, 2017 16:53:28 GMT
Gary I have been waiting for you to have a go at exercise 25. I haven't answered it myself because I too am convinced there is an error in the statement of the construction. I tried various changes to the instructions on how to perform the construction, but I couldn't get anything sensible out of it. I didn't mention that I thought there was an error to avoid giving you any preconceived ideas when you approached the exercise. I have spent a lot of effort on it, and I even sent an email to Needham at his university the University of San Francisco to ask him if he agreed that there is an error in the statement of the exercise, but just as he did last time I tried to contact him, he has ignored my email. I suppose it makes sense for him to do this, otherwise he would be inundated with emails about various things in his book. If I were working at a university and wrote to him in that capacity I suppose I might get a reply. I do know a few academics I could ask to do it on my behalf. Anyway I have given up on that possibility for the moment. I have looked at your solution. Here are some comments/questions: 1. From the sense of the exercise as a whole I think the asymptotics referred to are meant to be the asymptotics of $L$, not the h-line through $q'$. 2. Your construction does not really make use of the two h-circles (except perhaps $C'$ to determime $q'$ - but then what's special about $q'$?). We could just draw any h-line and choose a point not on this h-line and then construct the asymptotics to the h-line through the chosen point $p$. Here is a link to a diagram I made a while ago. It is very similar to yours. I have drawn a dashed circle and a dashed vertical line to represent the asymptotics to $L$ through $p$ (see figure 25 on page 305). Notice that they intersect $C$ in four points. I tried to find a construction that would intersect $C$ at two of these points ($a$ and $b$), one on each asymptotic, and then we could draw the asymptotics through these points and $p$ as it says in the exercise, but in the end I gave up. I must say I am feeling frustrated at not being able to find the error in the exercise statement. Do you remember the error we found in exercise 11 chapter 3 on page 183, that was also a construction exercise? Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Feb 15, 2017 5:12:04 GMT
Gary I have been waiting for you to have a go at exercise 25. I haven't answered it myself because I too am convinced there is an error in the statement of the construction. I tried various changes to the instructions on how to perform the construction, but I couldn't get anything sensible out of it. I didn't mention that I thought there was an error to avoid giving you any preconceived ideas when you approached the exercise. I have spent a lot of effort on it, and I even sent an email to Needham at his university the University of San Francisco to ask him if he agreed that there is an error in the statement of the exercise, but just as he did last time I tried to contact him, he has ignored my email. I suppose it makes sense for him to do this, otherwise he would be inundated with emails about various things in his book. If I were working at a university and wrote to him in that capacity I suppose I might get a reply. I do know a few academics I could ask to do it on my behalf. Anyway I have given up on that possibility for the moment. I have looked at your solution. Here are some comments/questions: 1. From the sense of the exercise as a whole I think the asymptotics referred to are meant to be the asymptotics of $L$, not the h-line through $q'$. 2. Your construction does not really make use of the two h-circles (except perhaps $C'$ to determime $q'$ - but then what's special about $q'$?). We could just draw any h-line and choose a point not on this h-line and then construct the asymptotics to the h-line through the chosen point $p$. Here is a link to a diagram I made a while ago. It is very similar to yours. I have drawn a dashed circle and a dashed vertical line to represent the asymptotics to $L$ through $p$ (see figure 25 on page 305). Notice that they intersect $C$ in four points. I tried to find a construction that would intersect $C$ at two of these points ($a$ and $b$), one on each asymptotic, and then we could draw the asymptotics through these points and $p$ as it says in the exercise, but in the end I gave up. I must say I am feeling frustrated at not being able to find the error in the exercise statement. Do you remember the error we found in exercise 11 chapter 3 on page 183, that was also a construction exercise? Vasco Vasco, 1. Then one of the asymptotics of L would be the vertical line through p and Re(p), as you have it. But as you argued with regard to q', what's special about these particular asymptotics that requires the construction of C and C'? And where are a and b? 2. What's special about q'? I take your point. I tried to conjure an answer to that in the last paragraph of my answer, but I didn't manage to convince myself that I was seeing a significant relationship. When he writes "the two asymptotic lines", that is a little vague, but it would normally apply to L, so I can see why you think it does not apply to the h-line through q' and why you drew them as you did. Yes, I took another look at the earlier exercise. Such problems are bound to occur. It's always interesting to hear what authors have to say about their own work. This is not a textbook for a mass readership, so a few replies would not be that time consuming, but perhaps Needham has moved on to other topics which command his attention. At least you gave him the opportunity. There are plenty of other problems to keep one occupied and a solution may occur to one of us in time. Even in the absence of a solution, I learned a couple of things from rereading related material and going through the construction. Gary |
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Post by Admin on Feb 15, 2017 5:30:00 GMT
Gary
Thanks. i have moved on to chapter 7 now, which is already giving me plenty to think about!
Vasco
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Post by telemeter on May 1, 2020 15:40:55 GMT
I promised myself I would come back to this, but to no avail. It seems so wrong I cannot even see how to fix it. Attached is a hyperbolic half-plane diagram like those above but with the asymptotes drawn in illustrating that there is no h-line orthogonal to L that passes through a and b. The question seems based on [25] p305...not that that helps. drive.google.com/file/d/1TPYqy1o88W5-zEdUXsXtdaWcf6mv7Pf0/view?usp=sharingUpdate! However, a,b and q' are collinear as per this drive.google.com/file/d/1tPN768EZd_qUp8i18TZ0IUu0zA53BoBM/view?usp=sharingIn this illustration it is clear that the h-line from the bottom/ top of one h-circle to the other does intersect the asymptotes at a and b (a' and b'). This indicates the question could be reworded as follows: Through q' draw the h-line that intersects C vertically above/below p. Call the intersections a and b. Show that the h-lines joining p to a and b are the two asymptotes! My approach on "showing" the required result is as attached drive.google.com/open?id=1P46eq1YNt6eokL1x5tzWuMsuZtZAbdkwtelemeter |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on May 11, 2020 20:21:59 GMT
I promised myself I would come back to this, but to no avail. It seems so wrong I cannot even see how to fix it. Attached is a hyperbolic half-plane diagram like those above but with the asymptotes drawn in illustrating that there is no h-line orthogonal to L that passes through a and b. The question seems based on [25] p305...not that that helps. drive.google.com/file/d/1TPYqy1o88W5-zEdUXsXtdaWcf6mv7Pf0/view?usp=sharingUpdate! However, a,b and q' are collinear as per this drive.google.com/file/d/1tPN768EZd_qUp8i18TZ0IUu0zA53BoBM/view?usp=sharingIn this illustration it is clear that the h-line from the bottom/ top of one h-circle to the other does intersect the asymptotes at a and b (a' and b'). This indicates the question could be reworded as follows: Through q' draw the h-line that intersects C vertically above/below p. Call the intersections a and b. Show that the h-lines joining p to a and b are the two asymptotes! My approach on "showing" the required result is as attached drive.google.com/open?id=1P46eq1YNt6eokL1x5tzWuMsuZtZAbdkwtelemeter telemeter, I have gone through your last document attempting to plot it step by step. I think your answer is likely the one intended by Needham, but I would still maintain that the exercise statement in the text needs a rewrite. On Line 3 of the exercise we find: “Through q' draw the h-line orthogonal to L, cutting C at a and b.” This is not generally possible. That is why I decided to replace “C” with the “horizon” in my own answer, for which the asymptotes are shown in gray. But it will also work if “orthogonal” is replaced with “asymptotic” (and the line is not drawn through q'). The bold lines show my representation of your answer. I made a small change in notation, swapping $a$ and $a^\prime$ so that the circular asymptotic h-line cuts c at a and b to conform more closely to the exercise statement. So the lower intersection of the half-line and C is labeled $a^\prime$.
Gary |
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