How to plot a circle on a spherical surface?

I have written a code which plots a circle(with center (x,y) and radius r) on the surface of a sphere. But, it does not work properly in the boundary, that is near y=0. The code:

close all; clear all; clc
N=100;
[X,Y,Z]=sphere(N);
C=zeros(N+1,N+1);

x=75;
y=1;
r=15;

for i=1:N+1
    for j=1:N+1
        d=sqrt(((i-x)^2)+((j-y)^2));
        if (d<=r)
            C(i,j)=1;
        end
    end
end

figure
surf(X,Y,Z,C)
axis equal
% shading interp
% light

The code works properly for values of y>r like y=15,16,17... Can anybody please help to make it work for values of y less than r?

[KALYAN ACHARJYA]

Check it for y=100, not circle. Also, if condition always false, by varying y value.

d<=r

Answer 1

[TED MOSBY]

Hi @Dhritiman Talukdar,

I understand that you want to plot a circle on the surface of the sphere. The issue you're encountering is related to how the circle is defined on the surface of the sphere. When y is close to zero, the circle might be partially or entirely outside the bounds of the sphere's surface parameterization due to the way you're calculating the distance d in a Cartesian coordinate system rather than on the sphere's surface.

Have a look at the below code for a possible solution:

close all; clear all; clc 
N = 100; 
[X, Y, Z] = sphere(N); 

% Parameters for the circle 
x = 75; % Longitude in degrees 
y = 1;  % Latitude in degrees 
r = 15; % Radius in degrees 

% Convert degrees to radians for computation 
x_rad = deg2rad(x); 
y_rad = deg2rad(y); 
r_rad = deg2rad(r); 

% Calculate the circle's center in Cartesian coordinates 
center_x = cos(y_rad) * cos(x_rad); 
center_y = cos(y_rad) * sin(x_rad); 
center_z = sin(y_rad); 

% Create a circle in spherical coordinates 
theta = linspace(0, 2*pi, 100); 
circle_x = cos(r_rad) * cos(theta); 
circle_y = cos(r_rad) * sin(theta); 
circle_z = sin(r_rad) * ones(size(theta)); 

% Rotate circle to the correct position on the sphere 
% Rotation matrix to align the circle with the sphere's surface 
R = [cos(x_rad), -sin(x_rad), 0; sin(x_rad), cos(x_rad), 0; 0, 0, 1] * ... 
    [cos(y_rad), 0, sin(y_rad); 0, 1, 0; -sin(y_rad), 0, cos(y_rad)]; 
circle_points = R * [circle_x; circle_y; circle_z]; 

% Plot the sphere 
figure 
surf(X, Y, Z, 'FaceAlpha', 0.5, 'EdgeColor', 'none'); 
hold on 

% Plot the circle on the sphere 
plot3(circle_points(1, :), circle_points(2, :), circle_points(3, :), 'r-', 'LineWidth', 2); 

% Set the view and other plot properties 
axis equal 
xlabel('X') 
ylabel('Y') 
zlabel('Z') 
title('Circle on Sphere') 
grid on 
hold off 

Here is the output of the above code in my MATLAB R2023b:

To get more information on the plot functions of MATLAB, have a look at the following documentation:

www.mathworks.com/help/matlab/surface-and-mesh-plots-1.html

www.mathworks.com/help/matlab/ref/plot3.html?searchHighlight=plot3&s_tid=srchtitle_support_results_1_plot3

Hope this resolves the problem!