Intersection of ellipsoid and cube

I need to determine the intersection of volume of ellipsoid with that of a cuboid in matlab. Any suggestions ?

[Walter Roberson]

Do you need only the volume, or do you need the coordinates of intersection as well?

[Sakshi]

I'm sorry for not seeing your reply. I only need the volume of intersection, no coordinates are required.

Answer 1

[Naga]

If you're looking to find the volume where an ellipsoid intersects with a cube, you can use a Monte Carlo simulation for a neat approximation.

• Use a million random points (1e6) for better accuracy.
• Generate random points within the cube's bounding box. For each point, check if it’s inside the ellipsoid using the equation ((x/a)^2 + (y/b)^2 + (z/c)^2 \leq 1).
• Estimate the intersection volume by multiplying the cube's volume by the ratio of points inside the ellipsoid to the total number of points.

Here's a snippet of code to do this:

a = 3; b = 2; c = 1; % Define the parameters of the ellipsoid
cube_center = [0, 0, 0]; cube_side = 4; % Define the parameters of the cube

% Define the number of random points for the Monte Carlo simulation
num_points = 1e6; % Increase this number for higher accuracy

% Generate random points within the bounding box of the cube
x = (rand(num_points, 1) - 0.5) * cube_side + cube_center(1);
y = (rand(num_points, 1) - 0.5) * cube_side + cube_center(2);
z = (rand(num_points, 1) - 0.5) * cube_side + cube_center(3);

% Check which points are inside the ellipsoid
inside_ellipsoid = (x.^2 / a^2) + (y.^2 / b^2) + (z.^2 / c^2) <= 1;

% Estimate the volume of the intersection
volume_cube = cube_side^3;
volume_intersection = volume_cube * sum(inside_ellipsoid) / num_points;

% Display the result
fprintf('Estimated volume of intersection: %.4f\n', volume_intersection);

Estimated volume of intersection: 21.4030

In my case, I got an estimated intersection volume of about 21.4030. If you want more accurate results, just increase the number of points.