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Post by mondo on Mar 20, 2023 1:04:52 GMT
This chapter starts of with an equation $H\{z+dz, z\} = \frac{|dz|}{Imz}$ how was the right hand side obtained? It should be a formula for the distance between any two points on the pseudosphere. Similarly on page 302 author proves the unique, shortest route between two veristically separated points by $d\hat{s} = \frac{ds_1}{y} < \frac{ds_2}{y} = d\hat{s_2}$ how are we sure the left hand side of this inequality is smaller than the RHS?
Additionally, I don't get this "The absence of parallel lines on a sphere implies that the direct motions can only be rotations." - Two questions to that. First, why there are no parallel lines on a sphere, what about latitudes each of them is parallel to the other right? Next, assuming there are no parallel lines as author claims, then why a direct motion can only be a rotation?
Thank you.
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Post by Admin on Mar 20, 2023 19:12:32 GMT
Mondo This chapter starts of with an equation $H\{z+dz, z\} = \frac{|dz|}{Imz}$ how was the right hand side obtained? It should be a formula for the distance between any two points on the pseudosphere. Similarly on page 302 author proves the unique, shortest route between two veristically separated points by $d\hat{s} = \frac{ds_1}{y} < \frac{ds_2}{y} = d\hat{s_2}$ how are we sure the left hand side of this inequality is smaller than the RHS? The answers to these two questions are so obvious that I am going to ask you to think again! Circles of latitude are not lines on the sphere. What is the definition of a line? Please can you say where your quotation "The absence...." can be found. Whenever you quote from the book please say where it can be found. Vasco
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Post by mondo on May 20, 2023 21:36:29 GMT
This chapter starts of with an equation $H\{z+dz, z\} = \frac{|dz|}{Imz}$ how was the right hand side obtained? It should be a formula for the distance between any two points on the pseudosphere. Similarly on page 302 author proves the unique, shortest route between two veristically separated points by $d\hat{s} = \frac{ds_1}{y} < \frac{ds_2}{y} = d\hat{s_2}$ how are we sure the left hand side of this inequality is smaller than the RHS? The answers to these two questions are so obvious that I am going to ask you to think again! Ok so, $H\{z+dz, z\} = \frac{|dz|}{Imz}$ appears to be just a substitution for $ds$ and $y$ in $d\hat{s} = \frac{ds}{y}$, am I right? For the second question on $d\hat{s} = \frac{ds_1}{y} < \frac{ds_2}{y} = d\hat{s_2}$ I am still not sure what makes us sure $ds_2 > ds_1$ - more so I think this is what we need to prove in order to conclude the shortest route theorem in (34). Is it because $ds_2$ is at an angle on figure [22a] and hence it must be longer? Circles of latitude are not lines on the sphere. What is the definition of a line? Please can you say where your quotation "The absence...." can be found. Whenever you quote from the book please say where it can be found. Vasco The simplest definition of a line is "In geometry, a line is an infinitely long object with no width, depth, or curvature." But in light of that there is no line at all on the sphere or pseudosphere as they cannot continue indefinitely and hence the "infinite" requirement is not met. Correct? The quotation "The absence...." is on page 301, 6th line below subchapter 5. So returning to my original question, why direct motions can only be rotations and not say translations or reflections?
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Post by mondo on May 21, 2023 6:12:59 GMT
I also don't understand equation (35) on page 302, how was the right hand side simplified to $|\ln{y_1/y_2}|$. I see we calculate the H-distance between two points that are separated vertically but I can't get this logarithm from this conclusion.
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Post by Admin on May 21, 2023 8:44:04 GMT
Mondo The answers to these two questions are so obvious that I am going to ask you to think again! Ok so, $H\{z+dz, z\} = \frac{|dz|}{Imz}$ appears to be just a substitution for $ds$ and $y$ in $d\hat{s} = \frac{ds}{y}$, am I right? For the second question on $d\hat{s} = \frac{ds_1}{y} < \frac{ds_2}{y} = d\hat{s_2}$ I am still not sure what makes us sure $ds_2 > ds_1$ - more so I think this is what we need to prove in order to conclude the shortest route theorem in (34). Is it because $ds_2$ is at an angle on figure [22a] and hence it must be longer? YesCircles of latitude are not lines on the sphere. What is the definition of a line? Please can you say where your quotation "The absence...." can be found. Whenever you quote from the book please say where it can be found. Vasco The simplest definition of a line is "In geometry, a line is an infinitely long object with no width, depth, or curvature." But in light of that there is no line at all on the sphere or pseudosphere as they cannot continue indefinitely and hence the "infinite" requirement is not met. Correct? yes, near the top of 269 Needham says we drop the requirement that lines be infinitely long.The quotation "The absence...." is on page 301, 6th line below subchapter 5. So returning to my original question, why direct motions can only be rotations and not say translations or reflections? Read subsection 3 pages 37-39 again to see the answer to your question Straight lines are also defined as the shortest distance between two points and the only lines on the sphere between two points which give the shortest distance are great circles. Circles of latitude are not great circles. The only lines on a sphere are great circles.Vasco
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Post by Admin on May 21, 2023 9:10:39 GMT
I also don't understand equation (35) on page 302, how was the right hand side simplified to $|\ln{y_1/y_2}|$. I see we calculate the H-distance between two points that are separated vertically but I can't get this logarithm from this conclusion. You need to integrate: $\displaystyle H\{z_1,z_2\}=H\{(x+iy_1),(x+iy_2)\}=\bigg|\int^{z_1}_{z_2}d\hat{s}\bigg|=\bigg|\int^{z_1}_{z_2}\frac{ds}{y}\bigg|=\bigg|\int^{z_1}_{z_2}\frac{dz}{y}\bigg|=\bigg|\int^{y_1}_{y_2}\frac{dy}{y}\bigg|$ Vasco
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Post by mondo on May 21, 2023 21:37:22 GMT
Thank you Vasco, it all makes sense now. The page 303 is all clear to me as well, actually I like the trick to prove $L$ is the shortest route from $z$ to $\tilde{z}$. However page 304 is quite difficult to follow. I need to reread it again but: 1. The reflection $R_K(z)$ is defined to be the same distance from $m$ but from figure [23b] we clearly see this is not the case and $[m\tilde{z}]$ is much longer than $[zm]$. 2. Isn't $\tilde{z}$ just the inversion point of $z$? The symbols follows what was done for inversion in previous chapters of the book. 3. The last paragraph of page 304 seems to be completely detached - there is a point $p$ that is said has infinitely many -h-lines through it that do not touch $L$ but how is that related to earlier paragraphs? What is this $p$ point? Why would h-lines through it touch $L$?
Thank you
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Post by Admin on May 21, 2023 22:10:40 GMT
Mondo
1. The h-distances are equal not the Euclidean distances in figure 23
Vasco
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Post by mondo on May 22, 2023 6:04:14 GMT
Ok. I stabled upon few more unclear things: 1. This sentence doesn't make sense to me (page 305, end of third paragraph) - "The angle of parallelism...tells you how far you can rotate $M$ before it starts missing $L$ entirely". The question is how does $\Pi$ tells that? What angle would indicate we no longer cross $L$? 2. Last paragraph of page 305, "In Euclidean geometry the analogue of two asymptotic lines is the unique parallel line through $p$, and since this is perpendicular to $M$ the analogue of $\Pi$ is a right angle." I can't imagine that, I try to use figure [24] but how can I have an asymptotic line through $p$ that is also perpendicular to $M$? I think an asymptotes shall be a tangent to $M$ right?
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Post by Admin on May 22, 2023 11:06:56 GMT
Ok. I stabled upon few more unclear things: 1. This sentence doesn't make sense to me (page 305, end of third paragraph) - "The angle of parallelism...tells you how far you can rotate $M$ before it starts missing $L$ entirely". The question is how does $\Pi$ tells that? What angle would indicate we no longer cross $L$? This is easy to see if you think about it and study figure 25. I'll leave it to you. I think there is an error in the book here. I think it should read: "In Euclidean geometry the analogue of the two asymptotic lines is the unique parallel line through $p$, and since this is parallel to $L$, the analogue of $\Pi$ is a right angle." Vasco
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Post by mondo on May 23, 2023 6:35:37 GMT
Ok. I stabled upon few more unclear things: 1. This sentence doesn't make sense to me (page 305, end of third paragraph) - "The angle of parallelism...tells you how far you can rotate $M$ before it starts missing $L$ entirely". The question is how does $\Pi$ tells that? What angle would indicate we no longer cross $L$? This is easy to see if you think about it and study figure 25. I'll leave it to you. I see the movement but I can't tell what would be the limiting value of $\Pi$ that cause lose of connection between $M$ and $L$ - can even tell what that angle would be? Or here it is just important that this angle will indeed tell that but the exact value of that angle cannot be calculated? 2. Last paragraph of page 305, "In Euclidean geometry the analogue of THE two asymptotic lines is the unique parallel line through $p$, and since this is perpendicular to $M$, the analogue of $\Pi$ is a right angle." I can't imagine that, I try to use figure [24] but how can I have an asymptotic line through $p$ that is also perpendicular to $M$? I think an asymptotes shall be a tangent to $M$ right? I think there is an error in the book here. I think it should read: "In Euclidean geometry the analogue of the two asymptotic lines is the unique parallel line through $p$, and since this is parallel to $L$, the analogue of $\Pi$ is a right angle." I have to reread it again, I have a feeling that there is no mistake in the book and we both miss something here.
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Post by Admin on May 23, 2023 14:15:40 GMT
Mondo
1. Once you rotate $M$ in figure 24 beyond either of the the asymptotic lines it no longer intersects $L$.
2. Have a good read.
Vasco
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Post by mondo on May 25, 2023 5:35:15 GMT
Mondo 1. Once you rotate $M$ in figure 24 beyond either of the the asymptotic lines it no longer intersects $L$. 2. Have a good read. Vasco Ok, I reread it again and here is how I understand it now: 1. I think this is really simple - $\Pi \neq 0$ means $L$ is crossing $M$ otherwise we lost it. 2. I agree with your correction to the book. However, the conclusion that the Euclidean analogy to $\Pi$ is a right angle comes from the fact that $L$ is perpendicular to $M$ and hence a parallel asymptotic must also make a right angle with $M$? Additionally there is one more quote that doesn't make sense -Last paragraph of page 305 - "...in hyperbolic geometry it is clear rgar $\Pi$ is always acute, and that its value decreases as the distance $D = H{p,q}$" of $p$ from $L$ increases. But when I try to visualize that on figure [25] it doesn't work this way. Let's imagine we rotate $M$ around $p$ counterclockwise. This means point $q$ will go up on $L$ and hence angle $\Pi$ will increase between asymptotic to $L$ and $M$. This contradicts this quote from the book.
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Post by Admin on May 25, 2023 11:50:11 GMT
Mondo
Referring to figure 25:
As point $q$ moves upwards then $L$ and $M$ must remain at right angles and so if we rotate $M$ anticlockwise about $p$ then $L$ will have to move towards the right, but remain a vertical line, and this means that the asymptotic to $L$ will also change shape at $p$ so that the angle $\Pi$ will become smaller.
Vasco
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Post by mondo on May 25, 2023 18:44:18 GMT
Mondo Referring to figure 25: As point $q$ moves upwards then $L$ and $M$ must remain at right angles and so if we rotate $M$ anticlockwise about $p$ then $L$ will have to move towards the right, but remain a vertical line, and this means that the asymptotic to $L$ will also change shape at $p$ so that the angle $\Pi$ will become smaller. Vasco Ok, if $L$ and its asymptotic are allowed to change the shape/positions then I think it makes sense. Do you agree with my conclusion in point 2. in post #12 about the right angle between $L$ and $M$? Thank you.
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