Plot 3d implicit function and inequality

Hi,

I need to plot a 3d equation and inequality, with some constraints. The equation and inequality are the following

(u.^2).*(p.^2).*(1-exp(-r.*z))-2.*u.*p.*t.*(1+exp(-r.*z))+(t.^2).*(1-exp(-r.*z))+4.*exp(-r.*z)=0
and 
(u.^2).*(p.^2).*(1-exp(-r.*z))-2.*u.*p.*t.*(1+exp(-r.*z))+(t.^2).*(1-exp(-r.*z))+4.*exp(-r.*z)<=0

r and t are scalars, say r=0.001 and t=1. u and z can take non-negative values, 0<=u<=1, and 0<=t, while p takes values that either satisfy the equation or inequality. In the case of the equality, I would like to have a 3d plot with u and z in the "lower" axis and p in the "vertical" axis. The same for the inequality though in this case I would need to represent the whole area that satisfies the inequality.

Thanks

Answer 1

[Sanchari]

Hello Fernando,

To plot the 3D equation and inequality in MATLAB, you can use the "fimplicit3" function for the equation and the "isosurface" function for the inequality. Below is a step-by-step guide to achieve this.

Step 1: define the equation and inequality as anonymous functions.

r = 0.001;
t = 1;

eqn = @(u, p, z) (u.^2).*(p.^2).*(1-exp(-r.*z))-2.*u.*p.*t.*(1+exp(-r.*z))+(t.^2).*(1-exp(-r.*z))+4.*exp(-r.*z);
ineq = @(u, p, z) (u.^2).*(p.^2).*(1-exp(-r.*z))-2.*u.*p.*t.*(1+exp(-r.*z))+(t.^2).*(1-exp(-r.*z))+4.*exp(-r.*z);

Step 2: To plot the equation, use fimplicit3 which plots the implicit function in 3D.

% Define the range for u, p, and z
u_range = linspace(0, 1, 100);
z_range = linspace(0, 10, 100); % Adjust range for z as needed
p_range = linspace(-10, 10, 100); % Adjust range for p as needed

% Plot the equation
figure;
fimplicit3(eqn, [0 1 -10 10 0 10], 'EdgeColor', 'none');
xlabel('u');
ylabel('p');
zlabel('z');
title('3D Plot of the Equation');
grid on;

Step 3: For the inequality, you can use isosurface to plot the volume where the inequality is satisfied.

% Create a grid of points
[u, p, z] = ndgrid(u_range, p_range, z_range);

% Evaluate the inequality on the grid
ineq_values = ineq(u, p, z);

% Plot the inequality using isosurface
figure;
isosurface(u, p, z, ineq_values, 0);
xlabel('u');
ylabel('p');
zlabel('z');
title('3D Plot of the Inequality');
grid on;

Here is the full MATLAB script that combines the above steps:

r = 0.001;
t = 1;

eqn = @(u, p, z) (u.^2).*(p.^2).*(1-exp(-r.*z))-2.*u.*p.*t.*(1+exp(-r.*z))+(t.^2).*(1-exp(-r.*z))+4.*exp(-r.*z);
ineq = @(u, p, z) (u.^2).*(p.^2).*(1-exp(-r.*z))-2.*u.*p.*t.*(1+exp(-r.*z))+(t.^2).*(1-exp(-r.*z))+4.*exp(-r.*z);

% Define the range for u, p, and z
u_range = linspace(0, 1, 100);
z_range = linspace(0, 10, 100); % Adjust range for z as needed
p_range = linspace(-10, 10, 100); % Adjust range for p as needed

% Plot the equation
figure;
fimplicit3(eqn, [0 1 -10 10 0 10], 'EdgeColor', 'none');
xlabel('u');
ylabel('p');
zlabel('z');
title('3D Plot of the Equation');
grid on;

% Create a grid of points
[u, p, z] = ndgrid(u_range, p_range, z_range);

% Evaluate the inequality on the grid
ineq_values = ineq(u, p, z);

% Plot the inequality using isosurface
figure;
isosurface(u, p, z, ineq_values, 0);
xlabel('u');
ylabel('p');
zlabel('z');
title('3D Plot of the Inequality');
grid on;

Pease consider the following:

• Adjust the ranges for u, p, and z as needed to capture the relevant parts of the plot.
• The "fimplicit3" function is used to plot the implicit equation, while "isosurface" is used to visualize the volume where the inequality is satisfied.
• The "isosurface" function plots the surface where the inequality function is zero, which helps visualize the boundary of the region satisfying the inequality.

Hope this helps!