doug
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Post by doug on Sept 9, 2024 6:16:01 GMT
I got the book 9 months ago and have become hooked, but it is a struggle that I would have given up on if not for your solutions Vasco, so thx. If I study them hard and long enough, then I can do the problems on my own (!). In part iv of Ex 1.29, how do you get the t = 1 point for the Z spiral, with arbitrary a and b? I see how to get the t = 0 pt because you know W = 1 for t = 0, so you rotate this point ACW by phi and then contract by r on a ray towards the origin. But I don't know what W equals for t = 1, so this approach fails me.
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Post by Admin on Sept 10, 2024 7:38:20 GMT
Hi Doug
Good to know you find the forum useful.
On page 2 of my answer I write $Z(t)=re^{i\phi}e^{at}e^{ibt}$
As you say when $t=0$ this becomes $Z(0)=re^{i\phi}.1.1=re^{i\phi}$
So when $t=1$ we have $Z(1)=re^{i\phi}e^{a}e^{ib}=Z(0)e^{a}e^{ib}$
So $Z(1)=Z(0)e^{a}e^{ib}=Z(0)W(1)$
We can see from this that to obtain $Z(1)$ we must rotate $Z(0)$ through an angle equal to $b$ (multiply by $e^{ib}$) and multiply by $e^{a}$ to change its length from r to $re^{a}$. An exactly similar operation to what you did with $W(0)=1$ to get $Z(0)$.
Hope that makes sense. Let me know if you have any more questions.
This illustrates the general point that when we multiply a complex number by another complex number we are rotating it and changing its length.
Vasco
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doug
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Posts: 6
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Post by doug on Sept 12, 2024 4:56:38 GMT
I think I understand now, but really what I need is a math program that can draw both spirals on the same page after input of a and b, and then animate the parametrizations, so I can play around and get a feel. Can you recommend one that would be useful for this and other problems in the book? Tristan mentioned Mathematica? (I'm 63 and have never used a math program, maybe a little old to learn, but I can throw 17 hours a day at it being retired!) Some further questions: 1) In figure 1.16, it looks like a = 1/2 and b = 1; what values were used to make your drawing? It appears b = pi divided by 2 (or about so) in your drawing since Z(0) was rotated by about 90 degrees in producing Z(1); 2) If different values of a and b were used but kept in the same ratio, then the same 2 spirals would be produced, Z(0) and W(0) would be the same, but Z(1) and W(1) would be different. Is all this correct?
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Post by Admin on Sept 12, 2024 9:36:15 GMT
Hi Doug One of the best ways of drawing graphs is to use Desmos which is free. I can't remember what values I used for $a$ and $b$ in my graph but I think those values you quote are about right. Here is a link to a Desmos program I have written which you can save to your own free desmos account. www.desmos.com/calculator/ehrdvt3fcn You can play around with this and change the values of $a$,$b$, and $R$ to see what happens and make a moving graph. You can also print out a user guide, but the best way to learn how to use it is just to experiment with it. You should be able to answer all your questions using this. Please ask if you need more help with the maths or with Desmos. Vasco |
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doug
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Post by doug on Sept 13, 2024 4:00:48 GMT
Thx, the Desmos program is everything I imagined. I'm going to spend some time on the tutorials and examples to come up to speed.
Doug
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doug
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Posts: 6
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Post by doug on Sept 13, 2024 6:55:04 GMT
I can easily see now that if a and b are changed but kept in the same ratio, then the angle phi between W(t) and Z(t) is unchanged, and the spiral W(t) is unchanged, but the spiral Z(t) is different because the expansion factor is changed. And the speed of the parametrization scales with b.
Doug
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