# How can I plot a boundary between real and imaginary solutions in a 3D graph?

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Paul on 20 Jun 2021
Commented: Paul on 25 Jun 2021
Hi everyone,
I have a function of three variables and I would like to plot a 3D graph where you can see the boundary (a surface or any shape) that separates the real and imaginary solutions of the function for a certain input. How can I do it?
So far, I have the following code:
fun = @(psi,theta,x)sqrt(-1.0./((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2-sqrt((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2-((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).^2+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2).*(sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2-((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).*(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0)).^2.*((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2-sqrt((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2-((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).^2+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2).*(sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0)-((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).^2+((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).*(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0)).^2+1.0).*(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(9.0./5.0e1)-sqrt((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2-((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).^2+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2).*(sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).*(9.0./5.0e1)-((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).^2.*(9.0./5.0e1)+((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).*(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).*(9.0./5.0e1))./((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2-sqrt((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2-((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).^2+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2).*(sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2-((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).*(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0))-1.0./5.0)+(((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2-sqrt((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2-((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).^2+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2).*(sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0)-((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).^2+((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).*(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0)).*(sin(theta).*(1.0e1./9.0)-cos(theta).*sin(psi).*(1.0e1./9.0)+sqrt(-1.0./((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2-sqrt((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2-((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).^2+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2).*(sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2-((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).*(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0)).^2.*((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2-sqrt((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2-((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).^2+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2).*(sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0)-((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).^2+((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).*(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0)).^2+1.0)-1.0e1./9.0).*(9.0./5.0e1))./((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2-sqrt((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2-((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).^2+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2).*(sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2-((sin(theta).*(-1.0./5.0)+cos(theta).*sin(psi).*(1.0./5.0)+1.0./5.0).^2.*(2.5e1./9.0)+(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0).^2.*(2.5e1./9.0)+(cos(psi).*(3.0./2.0e1)-3.0./2.0e1).^2.*(2.5e1./9.0)).*(x+cos(theta).*(1.0./5.0)+sin(psi).*sin(theta).*(1.0./5.0)-1.0./5.0));
% Input values:
psi = linspace(-pi/6,pi/6,50); theta = linspace(-pi/6,pi/6,50);x = linspace(0,0.5,50);
[PSI,THETA,X] = meshgrid(psi,theta,x);
% Solution for a certain input:
solution = fun(PSI,THETA,X);
% Obtain only the coordinates of real solutions and assign value
% "0" to imaginary solutions and value "1" to real ones.
coord_1 = [];
coord_2 = [];
coord_3 = [];
l = 1;
for i=1:length(psi)
for j=1:length(theta)
for k=1:length(x)
if imag(solution(i,j,k)) == 0
solution(i,j,k) = 1;
coord_1(l) = PSI(i,j,k);
coord_2(l) = THETA(i,j,k);
coord_3(l) = X(i,j,k);
l=l+1;
else
solution(i,j,k) = 0;
end
end
end
end
scatter3(coord_1',coord_2',coord_3',2);
By boundary I mean something similar to plot only the outer points (connected each other) of the scatter3 graph.

Vimal Rathod on 23 Jun 2021
Hi,
If you just want to connect points you could use the surf function on the x,y,z values obtained at the end. If you want to create just a boundary ignoring the inner points you could use the boundary function.
Hope this helps!
Paul on 25 Jun 2021
Hi Vimal,
Thanks for your answer. I wonder if there is an "optimal way" (without checking all the coordinates) to find the x,y,z values corresponding to the boundary. Because, without these values I can not use surf or boundary functions.

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