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Post by Admin on Feb 20, 2017 15:17:05 GMT
Gary and I are of the opinion that this exercise contains an error because if you follow the instructions for the construction, it doesn't work out satisfactorily. We have both tried to resolve the problem but have not yet found a sensible modification. We will keep trying.
Vasco
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Post by telemeter on May 1, 2020 18:20:57 GMT
Vasco
See my post under Exercise 25...a small step forward perhaps.
telemeter
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Gary
GaryVasco
Posts: 3,352 
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Post by Gary on May 11, 2020 20:35:56 GMT
Vasco See my post under Exercise 25...a small step forward perhaps. telemeter telemeter's answer seems correct to me.
Gary
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Gary
GaryVasco
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Post by Gary on May 11, 2020 21:05:34 GMT
Vasco,
If telemeter agrees, I propose changing the exercise statement as follows: Beginning on line 3 of the exercise, replace “, and let $q^\prime$ be one of the intersection points with L. Through $q^\prime$ draw the h-line orthogonal to L, cutting C at a and b.” with “. Draw the h-line asymptotic to L, cutting C at a and b.”
This solution works well for the "cutting C at a and b", but a new problem emerges with the last sentence "Show that the h-lines joining p to a and b are the two asymptotic lines!" because the single h-line that cut C at a and b must now become the two asymptotic lines that join p to a and b. The sentence could be rewritten: "Show that the h-lines joining $p$ to $a$ and $b$ and joining $p$ to $a^\prime$ and $b^\prime$ are the two asymptotic lines!"
Gary
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Post by Admin on May 11, 2020 22:03:07 GMT
Gary and telemeter
I haven't looked at your ideas yet. I will try and look in the next few days.
Vasco
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on May 11, 2020 22:21:48 GMT
Vasco and telemeter,
Here is the full proposed rewrite:
Given a point $p$ not on the h-line $L$, draw an h-circle $C$ of h-radius $\rho$ centred at $p$. Draw the h-line $M$ orthogonal to $L$ through $p$, cutting $L$ in $q$. Draw an h-circle $C^\prime$ of h-radius $\rho$ centred at $q$. Draw the h-line asymptotic to $L$, cutting $C$ at $a$ and $b$. Show that the h-lines joining $p$ to $a$ and $b$ and joining $p$ to $a^\prime$ and $b^\prime$ are the two asymptotic lines! [Hint: Take $L$ to be a vertical line in the upper half-plane.] What happens if we perform this construction in the Euclidean plane?
Gary
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Post by telemeter on May 25, 2020 8:59:26 GMT
Hi Gary,
Sorry for the delay...been buried in learning about p-forms. One day is so like another in lock-down!
I am not sure your proposed wording quite works because there are many asymptotes to L that will cut the circle C at two places, so when you say "Draw the h-line asymptotic to L, cutting C at a and b." that will not be enough to ensure that precisely the asymptote that is orthogonal to C is picked out. Also we would need to define a' and b' in the question.
Sorry to be a bit negative...especially as I do not have an improvement to offer! Equally, I don't see how to nicely specify the h-lines through q' a and b as I tried to in my approach in the question thread. IF anything comes to me I'll post it.
telemeter
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on May 25, 2020 14:45:29 GMT
Hi Gary, Sorry for the delay...been buried in learning about p-forms. One day is so like another in lock-down! I am not sure your proposed wording quite works because there are many asymptotes to L that will cut the circle C at two places, so when you say "Draw the h-line asymptotic to L, cutting C at a and b." that will not be enough to ensure that precisely the asymptote that is orthogonal to C is picked out. Also we would need to define a' and b' in the question. Sorry to be a bit negative...especially as I do not have an improvement to offer! Equally, I don't see how to nicely specify the h-lines through q' a and b as I tried to in my approach in the question thread. IF anything comes to me I'll post it. telemeter telemeter, I take no offense at either the delay or the doubts. I find myself without enough time for all the math in view even on this forum alone. I thought it was understood that the h-line cutting C at a and b would also pass through p, but perhaps not. Would it help to specify that? Then only one h-line would qualify. One could write “Draw the h-line passing through p and cutting C at a and b”, omitting mention of a vertical asymptote. Then perhaps one could show that a and b have inverses at $a^\prime$ and $b^\prime$ on the vertical through p, showing that the two h-lines (the vertical and the line through a and b) are the two asymptotes. I think in this interpretation of the question, one would not require $C^\prime$ or $q^\prime$ at all. I had the impression that the original question was somehow a conflation of two separate or sequential problems. I saw your discussion of translating on M and found it quite interesting, but I wasn’t able to apply it to my understanding of the question. I thought perhaps it would provide the answer to the second, still unforrmed, question involving $C^\prime$ and $q^\prime$. Gary |
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