Draw and color the intersection between two 3D surfaces?

I want to draw, color, and shading (optional) the intersection between these two surfaces

• Surface 1:
• Surface 2:

I read many posts but I found none of them have the same idea I'm going for. I don't know exactly what to search for or what to read, everything seems a little confuse. Thanks.

[Star Strider]

This is either straightforward (if ‘x’ and ‘y’ are bounded to be between -1 and 1 and so can be expressed as trigonometric functions) or much more difficult (if they are unbounded and are allowed to be complex). In the first instance, ‘z’ is actually a volume (probably cylindrical) that extends from a (circular) plane equal to 1 to .

[Sejuani Sylaas]

Yes you are correct and I also want to confirm that x and y are bounded to be between -1 and 1.

This is my first time using MATLAB for this kind of surface drawing task. So, I'm curious if you can give me some guidance to solve this problem? I really don't know where to start.

Answer 1

[Star Strider]

In that event, the code would go something like this —

syms x y z theta 
S1 = z <= sqrt(x^2+y^2)

S1 = 

S2 = x^2 + y^2 + (z-1)^2 <= 1

S2 = 

S1 = subs(S1, {x,y},{cos(theta),sin(theta)})

S1 = 

S2 = subs(S2, {x,y},{cos(theta),sin(theta)})

S2 = 

S1 = simplify(S1, 500)

S1 = 

S2 = simplify(S2, 500)

S2 = 

th = linspace(0, 2*pi).';

figure
surf((cos(th)*[1 1]), (sin(th)*[1 1]), (ones(size(th))*[-1 1]), 'FaceColor',[1 1 1]*0.5)
hold on
patch(cos(th)*[-1 1], sin(th)*[-1 1], ones(size(th))*[-1 1], [1 1 1]*0.5)
hold off
axis('equal')
view(30,30)
zt = zticks;
zticklabels(["-\infty" string(zt(2:end))])

.

Answer 2

[Bruno Luong]

It's sphere extruded by a cone

x = linspace(-1.1,1.1,129);
y = linspace(-1.1,1.1,129);
z = linspace(0,1.1,129);
[X,Y,Z] = meshgrid(x,y,z);
V = Z <= sqrt(X.^2 + Y.^2) & ...
    X.^2 + Y.^2 + (Z-1).^2 <= 1;
isosurface(X, Y, Z, V, 0.5);
axis equal

[Bruno Luong]

Now that we confirm the shape of the intersection, we can plot the surface boundary of the intersection usng customized patch

n = 20; % discretization parameters of spherical parts
W = allVL1(3, n); % FEX file https://www.mathworks.com/matlabcentral/fileexchange/17818-all-permutations-of-integers-with-sum-criteria

% Connectivity
XY = W*[0 1 0;
        0 0 1].';
F=delaunay(XY);
XYZ = W/n;

XYZs = XYZ ./ sqrt(sum(XYZ.^2,2));
XYZc = XYZs;
XYZs(:,3) = 1-XYZs(:,3);
XYZc(:,3) = sqrt(sum(XYZc(:,1:2).^2,2));

close all
patcharg = {'FaceColor', 'interp', 'EdgeColor', 'none', 'FaceLighting', 'gouraud'};
T = [0 -1  0;
     1  0  0;
     0  0  1];
for i = 0:3
    patch('Faces', F, 'Vertices', XYZs, 'CData', XYZs(:,3), patcharg{:});
    XYZs = XYZs*T';
    patch('Faces', F, 'Vertices', XYZc, 'CData', XYZc(:,3), patcharg{:});
    XYZc = XYZc*T';
end

view(3)
axis equal

With lighting set with camera toolbar