Azzera filtri
Azzera filtri

How to plot these 3-d plots with different domains?

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simran
simran il 1 Ott 2023
Commentato: simran il 10 Dic 2023
I have this surface parametrization where z belongs to open unit disk. Now we substitute i.e., , then we have:
Now they say we are using right-half plane as domain (because of the substitution) instead of the standard unit disk. can you tell a bit how?
The main problem is that they have plotted the following:
How have they plotted both these surfaces? If someone could please help me this code? Also, why do they write Re(Z)<0 in the right image as caption? shouldn't it be Re(z)>0?

Risposte (1)

Rangesh
Rangesh il 7 Dic 2023
Modificato: Rangesh il 7 Dic 2023
Hi Simran,
I understand you want to know the reason behind the transformation of w and plot the surface parametrization of the equation using the transformation of w.
Regarding the transformation, I believe that the unit disk maps onto the left half of the plane . The proof is as follows:
We know . After substitution we get,
Now, Squaring both sides, we get
After simplification, we obtain
Whereas the range mentioned in the caption of the right image specifies the area over which the function is calculated but that doesn’t mention as the transformation to have a range of
To plot the function , you can check the following steps used:
1.Taking z belong to the ball of radius 1 centered around origin,
We generate the points as follows.
np=100; %number of points
theta=linspace(0,2*pi,np);
radius= linspace(0,1,np);
2.Create a mesh grid, which produces the polar coordinates of the disk.
[R,Angle] = meshgrid(radius,theta);
3.Now, Z corresponds to the location of point in a complex plane, and W is the transformation
Z=R.*(cos(Angle)+1i*sin(Angle));
W=(1+Z)./(1-Z);% applying transformation
4.Defining the function for each of the 3D coordinates using the function handle
xfunc=@(w) real(w-1./w)/4;
yfunc=@(w) imag (log (w))/2;
zfunc=@(w) imag (w+ 1./w)/4;
5.Now, we obtain the coordinate by substituting the values into the functions and get the plot using surf function
surf(xfunc(W), yfunc(W), zfunc(W));
6. To obtain the surface plot of a general surface D. You can generate the points corresponding to D following the steps from (1) to (3) to plot.
For understanding the function used, you check the following links:
I hope this resolves your query.
Thanks,
Rangesh.
  1 Commento
simran
simran il 10 Dic 2023
After you said " we know |z|<1, after substitution we get |w+1|<|w-1|",
shouldn't it be |w-1|<|w+1|?

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