How to bound at 3D surface within an ellipse?

Hello! I feel like I looked everywhere for my answer but I could not find a solution, although it may be a simple one. I am relatively new to MATLAB. Basically I have a hyperbolic surface that I want to 3d plot. However, I only want to plot the part of the surface that is within an ellipse. The other boundaries for the hyperbolic surface are written below. Is there a function that can do this for me? Thank you!

clc;
clear all;

%defining constants 
E0= 22.232;
Emajor=22.032;
Eminor=15.574;
H0= 0.230;
syms x y z Hmajor(z) Hminor(z)

%First equation of the ellipse
eqn1 = ((x-(E0-28))/Emajor)^2 + ((z)/Eminor)^2 == 1;
fimplicit3(eqn1); %plot the ellipse
xlabel('X axis')
ylabel('Y axis')
hold on;

Hmajor(z) = E0 - (Emajor * (1 - (z/Eminor)^2)^0.5) - H0; 
Hminor(z) = Hmajor(z) * tand(60);

% Second equation (hyperbola) using Hmajor(z) and Hminor(z)
eqn2 = ((x-H0)/Hmajor(z))^2 - (y/Hminor(z))^2 == 1;
interval=[0 50 -25 25 -20 0];
fimplicit3(eqn2,interval) %plot hyperbola over the ellipse plot

axis equal
xlabel('X axis')
ylabel('Y axis')
zlabel('Z axis')

Answer 1

[Zinea]

Hi Hope,

I understand that the goal is to plot a hyperbolic surface constrained within an elliptical region. This can be achieved by evaluating the hyperbola only where it satisfies the ellipse equation. The following are the steps to be followed:

1. A 3D grid is created using “meshgrid” to evaluate the equations over a specified range.

[xGrid, yGrid, zGrid] = meshgrid(linspace(-25, 25, 50), linspace(-25, 25, 50), linspace(-20, 0, 50));

The following documentation link of “meshgrid” may be helpful:

https://www.mathworks.com/help/matlab/ref/meshgrid.html

2. A logical mask “ellipseMask” is calculated to determine which points lie inside the ellipse.

ellipseMask = ((xGrid-(E0-28))/Emajor).^2 + (zGrid/Eminor).^2; 

3. The hyperbola equation is evaluated over the grid and the mask “ellipseMask” is applied.

hyperbolaValues = real(((xGrid-H0)./Hmajor(zGrid)).^2 - (yGrid./Hminor(zGrid)).^2); 

4. The full hyperbolic surface is plotted using “isosurface”.

isosurface(xGrid, yGrid, zGrid, hyperbolaValues, 0);

You may refer to the documentation of “isosurface” to know how to use it:

https://www.mathworks.com/help/matlab/ref/isosurface.html

5. The ellipse is plotted similarly as above.

isosurface(xGrid, yGrid, zGrid, ellipseMask, 1); 

6. The hyperbolic surface within the ellipse is calculated as below:

ellipseMask = ellipseMask <= 1; 
constrainedHyperbolaValues = hyperbolaValues; 
constrainedHyperbolaValues(~ellipseMask) = NaN; 

7. Finally, the hyperbolic surface calculated above is plotted using “isosurface”.

isosurface(xGrid, yGrid, zGrid, constrainedHyperbolaValues, 0); 

Given below are the figures for the full hyperbolic surface (Fig 1), ellipse (Fig 2) and the hyperbolic surface within the ellipse (Fig 3) respectively:

Fig 1

Fig 2

Fig 3

[Hope]

Thank you so much! This was very helpful!