MATLAB: How to rotate 3D cube so that one face is normal to a given axis

matrix manipulationrotation

Hello everyone,
I am havign a bit of a headache and am a little stuck. I would really appreciate any help.
I have some 3D points (in a cube shape shown as blue in the figure). I would like to rotate these points such that one face of the cube is normal to a line passing through the origin of the system (red dot in the figure) and the centre of the cube. I have tried a few things but it doesn't quite work and some edge cases throw the system way off.
In the example below it looks about right but when you rotate the right hand figure you can see that the cube is not quite normal to the line.
Hopefully the script below will help explain what I mean. 
%% Variables for Matlab question
% Centre of system
    coords_source_xyz= [-0.5,-0.5,0; -0.5,0,0;-0.5,0.5,0;0,-0.5,0;0,0,0;0,0.5,0;0.5,-0.5,0;0.5,0,0;0.5,0.5,0];
    array_centre =[0,0,0];
% Grid of unrotated cloud points. 
    sample_grid_coords_x = linspace(2,3,10);
    sample_grid_coords_y = linspace(-2,-3,10);
    sample_grid_coords_z = linspace(2,3,10);
% Centre of un-rotated point cloud
    centre_grid_x = mean(sample_grid_coords_x);
    centre_grid_y = mean(sample_grid_coords_y);
    centre_grid_z = mean(sample_grid_coords_z);
% NDGRID OF points 
    [sample_X, sample_Y, sample_Z] = ndgrid(sample_grid_coords_x, sample_grid_coords_y, sample_grid_coords_z);
% Distance from centre of array to centre of unrotated point cloud. 
    array_to_grid_x = centre_grid_x-array_centre(1);
    array_to_grid_y = centre_grid_y-array_centre(2);
    array_to_grid_z = centre_grid_z-array_centre(3);
    array_to_grid_xy = sqrt((array_to_grid_x^2) + (array_to_grid_y^2));
    
% Move the point cloud back to the origin temporarily. 
    sample_X = sample_X - array_to_grid_x;
    sample_Y = sample_Y - array_to_grid_y;
    sample_Z = sample_Z - array_to_grid_z;
% Step 1 - Calculate rotation angle about Y. 
    rot_y_axis_deg = atand(array_to_grid_z / array_to_grid_x);
    rot_y_axis_deg(isnan(rot_y_axis_deg))=0;
% Rotate points about Y axis first. 
    X_rot_1 = sample_X*cosd(rot_y_axis_deg) + sample_Z*sind(rot_y_axis_deg);
    Y_rot_1 = sample_Y;
    Z_rot_1 = sample_Z*cosd(rot_y_axis_deg) - sample_X*sind(rot_y_axis_deg);
% Step 2 - Calculate rotation angle about Z. 
    rot_z_axis_deg =  atand(array_to_grid_x / array_to_grid_y);
    rot_z_axis_deg(isnan(rot_z_axis_deg))=0;
% Rotate points about Z axis second. 
    X_rot_2 = X_rot_1*cosd(rot_z_axis_deg) - Y_rot_1*sind(rot_z_axis_deg);
    Y_rot_2 = X_rot_1*sind(rot_z_axis_deg) + Y_rot_1*cosd(rot_z_axis_deg);
    Z_rot_2 = Z_rot_1;
% Move them back to their original centre
    X_rot_2 = X_rot_2+centre_grid_x;
    Y_rot_2 = Y_rot_2+centre_grid_y;
    Z_rot_2 = Z_rot_2+centre_grid_z;
figure(1)
    subplot(1,2,1); 
    % Location of array points

    scatter3(coords_source_xyz(:,1), coords_source_xyz(:,2), coords_source_xyz(:,3),'k', 'filled'); daspect([1 1 1]); title('Un-Rotated');  
    hold on
    % Location of centre of array

    scatter3(array_centre(1), array_centre(2), array_centre(3),'r', 'filled'); daspect([1 1 1]); 
    % Location of sample grid points - unrotated
    scatter3(sample_X(:)+array_to_grid_x, sample_Y(:)+array_to_grid_y, sample_Z(:)+array_to_grid_z, 'b'); 
    % Line connecting centre of array and centre of grid

    x = [array_centre(1),  centre_grid_x];y = [array_centre(2),  centre_grid_y]; z=[array_centre(3), centre_grid_z]
    plot3(x*1.5, y*1.5, z*1.5); hold off
    subplot(1,2,2); 
    % Location of array points
    scatter3(coords_source_xyz(:,1), coords_source_xyz(:,2), coords_source_xyz(:,3),'k', 'filled'); daspect([1 1 1]);  title('Rotated');
    hold on
    % Location of centre of array
    scatter3(array_centre(1), array_centre(2), array_centre(3),'r', 'filled'); daspect([1 1 1]); 
    % Location of sample grid points - rotated
    scatter3(X_rot_2(:), Y_rot_2(:), Z_rot_2(:), 'b'); 
    % Line connecting centre of array and centre of grid
    x = [array_centre(1),  centre_grid_x];y = [array_centre(2),  centre_grid_y]; z=[array_centre(3), centre_grid_z]
    plot3(x*1.5, y*1.5, z*1.5); hold off

Best Answer

If I can choose the missing rotation angle freely, then here is one solution.
% Centre of system
    coords_source_xyz= [-0.5,-0.5,0; -0.5,0,0;-0.5,0.5,0;0,-0.5,0;0,0,0;0,0.5,0;0.5,-0.5,0;0.5,0,0;0.5,0.5,0];
    array_centre =[0,0,0];
% Grid of unrotated cloud points. 
    sample_grid_coords_x = linspace(2,3,10);
    sample_grid_coords_y = linspace(-2,-3,10);
    sample_grid_coords_z = linspace(2,3,10);
    
    [sample_X, sample_Y, sample_Z] = ndgrid(sample_grid_coords_x, sample_grid_coords_y, sample_grid_coords_z);
   
    Cube=[sample_X(:), sample_Y(:), sample_Z(:)];
    
% Perform rotation
   cube_centre=mean(Cube,1);  %Cube center
   
   ray=cube_centre-array_centre;
   R=normalize([ray(:),null(ray)],1,'norm'); %rotation matrix
   
   C = num2cell( (Cube-cube_centre)*R.'+ cube_centre, 1);
   
   [X_rot, Y_rot, Z_rot] = deal(C{:});
   %%%% Plot %%%%
   
figure(1)
    % Location of array points
    scatter3(coords_source_xyz(:,1), coords_source_xyz(:,2), coords_source_xyz(:,3),'k', 'filled'); daspect([1 1 1]);  title('Rotated');
    hold on
    % Location of centre of array
     scatter3(array_centre(1), array_centre(2), array_centre(3),'r', 'filled'); daspect([1 1 1]); 
    % Location of sample grid points - rotated
    scatter3(X_rot, Y_rot, Z_rot, 'b'); 
    % Line connecting centre of array and centre of grid
    x = [array_centre(1),  cube_centre(1)];
    y = [array_centre(2),  cube_centre(2)]; 
    z=  [array_centre(3),  cube_centre(3)];
    plot3(x*1.5, y*1.5, z*1.5);
    hold off
Related Question