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

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?

Answer 1

[Rangesh]

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:

• “linspace” return a vector spaced between the points x1 and x2 : https://in.mathworks.com/help/matlab/ref/linspace.html
• “meshgrid” this return the 2D grid coordinate based on the coordinates of x and y: https://www.mathworks.com/help/matlab/ref/meshgrid.html
• Function Handle this creates a anonymous function with one line expression t o evaluate:https://in.mathworks.com/help/matlab/ref/function_handle.html
• “surf” which visualizes the surface plots: https://www.mathworks.com/help/matlab/ref/surf.html [HS3]

I hope this resolves your query.

Thanks,

Rangesh.

[simran]

After you said " we know |z|<1, after substitution we get |w+1|<|w-1|",

shouldn't it be |w-1|<|w+1|?