MATLAB - How to project an sphere/hemisphere to any planes?

I have objects as a figure attached. I want to project objects (blue color and small red color) to plane (yellow color) follow perpendicular.

I can project object to coordinate plane (oxy, oyz, oxz) by command: surf(x,y,0*z), surf(0*x,y,z), surf(x,0*y,z). How to project to any plane?

Thank

this is 2D projected figure when project to plan ozx </matlabcentral/answers/uploaded_files/58418/project%20to%20coordinate%20plane.png>

[viet le]

this is 2D projected figure when project to plan ozx

[Jianshe Feng]

Did you get this solved? I have the same question now.

Answer 1

[Aniket]

Hi @viet le and @Jianshe Feng,

To project 3D objects (such as your blue and red shapes) onto an arbitrary plane (like the yellow one shown in your figure) perpendicularly, you'll need to use a vector-based approach.

Assume the plane is defined by:

A point on the plane: P0 = [x0, y0, z0]

A normal vector: n = [A, B, C]

Then for any 3D point P = [x, y, z], the perpendicular projection onto the plane is computed as:

n = n / norm(n); % Normalize the normal vector
d = dot(n, (P - P0)); % Distance from point to plane along the normal
P_proj = P - d * n;  % Projected point on the plane

This can be done efficiently for an entire set of points representing your objects.

% Define the plane
plane_point = [0, 0, 0];         % A point on the plane
plane_normal = [1, 1, 1];        % Normal vector to the plane

% Sample object: a sphere
[x, y, z] = sphere(20);
x = x(:); y = y(:); z = z(:); % Convert to point list

% Combine points into a matrix
points = [x, y, z];

% Normalize normal vector
n = plane_normal / norm(plane_normal);

% Compute perpendicular projection for each point
distances = (points - plane_point) * n';
projected_points = points - distances .* n;

% Plot original and projected points
figure;
scatter3(points(:,1), points(:,2), points(:,3), 'b.'); hold on;
scatter3(projected_points(:,1), projected_points(:,2), projected_points(:,3), 'r.');
legend('Original Points', 'Projected Points');


% Optionally: Plot the projection plane
[xp, yp] = meshgrid(-2:0.5:2, -2:0.5:2);
zp = (-n(1)*(xp - plane_point(1)) - n(2)*(yp - plane_point(2))) / n(3) + plane_point(3);
surf(xp, yp, zp,'FaceAlpha', 0.3, 'EdgeColor', 'none', 'FaceColor', 'yellow');

axis equal;
title('Projection of 3D Objects onto Arbitrary Plane');

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If the plane is defined by an equation in the standard form: Ax+By+Cz+D=0, extract the normal vector from coefficients and follow the same math.

I hope this answers your query!