MATLAB: Draw a circle for arbitrary orientation on spherical surface

3d plots

I would like to draw a circular loop on spherical surface for a fixed orientation of theta(i.e. polar angle on sphere) and variying the azimuthal angle. The following code only generates the circle which are parallel to the equiatorial plane, but I need arbitrary orientation of ploar angle on the sphere. Pl somebody help me.
clear; clc;
N=10;
[X,Y,Z]=sphere(N);
C=zeros(N+1,N+1);
x=7;
y=1;
r=10;
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

Best Answer

My approach is to define the circle around the X-axis, then rotate it into the desired position through a Yaw-Pitch rotation.
% Specify the user inputs
Rmag= 10;   % Sphere radius
radius = 1; % circle radius
psi = 30*pi/180;    % yaw rotation angle
theta = -45*pi/180; % pitch rotation angle (negative rotation is up)
% Define vectors and Calculate the YAW-PITCH transformation matrix
alpha = asin(radius/Rmag);  
YAW = [cos(psi), -sin(psi), 0; sin(psi), cos(psi), 0; 0, 0, 1]; % Planar YAW rotation
PITCH = [cos(theta), 0, sin(theta); 0,1,0; -sin(theta), 0, cos(theta)]; % Planar PITCH rotation
YP = YAW*PITCH; % YAW-PITCH rotation matrix
% Rc = Column Vector pointing to circle on X-Axis (start point)
Rc = [Rmag*cos(alpha); 0; radius];  
R = YP*[Rmag; 0; 0]; % Vector pointing to Circle center 
% Now sweep the Rc vector around the X-axis to generate the circle 
% This is done by adding a planar ROLL rotation to YP
clear C
C = [];  % C is the vector containing the circle X-Y-Z cordinates
for phi = 0:pi/50:2*pi
    ROLL = [1, 0, 0;  0, cos(phi), -sin(phi); 0, sin(phi), cos(phi) ]; 
    YPR = YP*ROLL;  % 3-dimentional transform
    Rnew = YPR*Rc;
    C = [C, Rnew];
end
% plot the result
[Sx,Sy,Sz]=sphere(50);
figure;
mesh(Rmag.*Sx, Rmag.*Sy, Rmag.*Sz);  % draw the sphere
hold on;
plot3([0, R(1)], [0, R(2)], [0,R(3)], 'r'); % Vector to Circle center
plot3(C(1,:), C(2,:),C(3,:), 'b'); % Circle 
axis equal
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