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Post by mondo on May 29, 2023 20:47:01 GMT
In the fourth paragraph on page 349 author presents the local effect of a conjugation mapping - "For example, conjugation has (taking the first expansion to be horizontal) $\xi_a = 1, \eta_a = -1, \phi_a = 0$"
The way I understand it is, there is unity stretch in the horizontal direction (no change), negative unity stretch in a perpendicular direction and finally no rotation at all. The unity stretch in horizontal direction makes sense to me but the $-1$ in perpendicular direction doesn't. First of all conjugation does not necessary transform a given complex number into an image number that is $90$ degrees apart right? For instance $z = 3x + iy$ and $\bar{z} = 3x - iy$ are $36$ degrees apart. Hence these numbers here are only valid for this particular example $z = x + iy$
Even more important question is why don't we make use of a rotation? Here rotation $\phi_a = 0$. Why don't we say $\phi = -\frac{\pi}{2}$ instead?
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Post by Admin on Jun 2, 2023 11:58:30 GMT
In the fourth paragraph on page 349 author presents the local effect of a conjugation mapping - "For example, conjugation has (taking the first expansion to be horizontal) $\xi_a = 1, \eta_a = -1, \phi_a = 0$" The way I understand it is, there is unity stretch in the horizontal direction (no change), negative unity stretch in a perpendicular direction and finally no rotation at all. The unity stretch in horizontal direction makes sense to me but the $-1$ in perpendicular direction doesn't. First of all conjugation does not necessary transform a given complex number into an image number that is $90$ degrees apart right? For instance $z = 3x + iy$ and $\bar{z} = 3x - iy$ are $36$ degrees apart. Hence these numbers here are only valid for this particular example $z = x + iy$ Even more important question is why don't we make use of a rotation? Here rotation $\phi_a = 0$. Why don't we say $\phi = -\frac{\pi}{2}$ instead? Mondo Needham is talking here about local mappings as described in chapter 4. Think of a point at the centre of a small circle and how this circle is mapped. The ideas are similar to those used in the explanation of the amplitwist. You will need to read and understand the whole of subsection 3 on pages 349-350 and refer to chapter 4 if you need to. Vasco |
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Post by mondo on Jun 3, 2023 7:27:47 GMT
In the fourth paragraph on page 349 author presents the local effect of a conjugation mapping - "For example, conjugation has (taking the first expansion to be horizontal) $\xi_a = 1, \eta_a = -1, \phi_a = 0$" The way I understand it is, there is unity stretch in the horizontal direction (no change), negative unity stretch in a perpendicular direction and finally no rotation at all. The unity stretch in horizontal direction makes sense to me but the $-1$ in perpendicular direction doesn't. First of all conjugation does not necessary transform a given complex number into an image number that is $90$ degrees apart right? For instance $z = 3x + iy$ and $\bar{z} = 3x - iy$ are $36$ degrees apart. Hence these numbers here are only valid for this particular example $z = x + iy$ Even more important question is why don't we make use of a rotation? Here rotation $\phi_a = 0$. Why don't we say $\phi = -\frac{\pi}{2}$ instead? Mondo Needham is talking here about local mappings as described in chapter 4. Think of a point at the centre of a small circle and how this circle is mapped. The ideas are similar to those used in the explanation of the amplitwist. You will need to read and understand the whole of subsection 3 on pages 349-350 and refer to chapter 4 if you need to. Vasco Yes I read these pages 349-500 and it makes sense to me. The only doubt I have is why in the example he gave, which I quoted we don't use the rotation while we can? After all expansion in perpendicular direction by $1$ is equal to a rotation by $\pi/2$. That also makes me wonder when, under what circumstances we would use a rotation? Anyway I think this part is not that important to understand the rest of the chapter. |
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Post by Admin on Jun 3, 2023 7:46:45 GMT
Mondo
From what you have written above I think you do not understand what Needham is saying here. If you don't understand this then it means you probably don't really understand what the amplitwist is all about.
Vasco
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Post by Admin on Jun 3, 2023 11:22:37 GMT
Mondo
I don't understand why you talk about rotation through 90 degrees. Where does that idea come from?
In reply #2 you say "After all expansion in perpendicular direction by $1$ is equal to a rotation by $\pi/2$." I don't understand why you say this, can you explain it more clearly please?
If you draw a small circle $C_a$ above the real axis centred at $a=\overline{p}$, and a radius vector of $C_a$ from $a$ to any point on the circumference, and then apply the mapping $h(z)=\overline{z}$ to the plane, then the image circle will be centred at $p$ and the radius vector in the image circle will now go from $p$ to the image point of the tip of the preimage vector.
If we draw this diagram then we can see that the image radius vector is obtained exactly as described on page 349 paragraph 3:
The real component of the image radius vector is the same as the preimage, the imaginary component of the image radius vector is the same as the preimage with sign reversed and finally the radius vector is not rotated.
Because conjugation is not an analytic mapping it does not have an amplitwist at $a$. It has an amplification but not a twist, since we would have to rotate each radius vector by a different amount. See the paragraph on page 200 just above figure 13.
Vasco
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Post by mondo on Jun 3, 2023 15:24:58 GMT
Vasco,
thank you for this response, it is very clear. In my earlier post, when I said "After all expansion in perpendicular direction by 1 is equal to a rotation by $\pi/2$ I meant to say that if we are expanding in parapedicular direction we can as well rotate by $90$ degrees. However then have to do another perpendicular expansion to maintain the horizontal expansion so my proposal really doesn't make sense. Plus your argument with the non-analytic mapping is even stronger one.
Thanks again!
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