How to calculate the volume of a 3D triangular mesh?

I have a pyramid with these vertices (each side is a triangle):

[-4 0 5;
1 -5 5;
1 0 1;
1 5 5;
1 0 0]

How can I calculate it's volume?

I couldn't quite understand this answer.

[darova]

How does it look like?

[Adam Danz]

How is this a pyramid?

xyz = [-4 0 5;
    1 -5 5;
    1 0 1;
    1 5 5;
    1 0 0];

% Wrap last point to first point
xyz(end+1,:) = xyz(1,:);

%What's this look like?
clf()
plot3(xyz(:,1),xyz(:,2),xyz(:,3),'b-o')
text(xyz(:,1),xyz(:,2),xyz(:,3),{'1' '2' '3' '4' '5' ''},'VerticalAlignment','bottom')
% fill3(xyz(:,1),xyz(:,2),xyz(:,3),'b')
grid on
axis equal
xlabel('x axis')
ylabel('y axis')
zlabel('z axis')

[Hadi Ghahremannezhad]

It looks like this:

It's a whole framework for triangular mesh modeling.

The specific part for calculating the volume is like this:

function vol = computeVolume(mesh)
            % should retrun the signed volume of the mesh.

            % Compute the signed volume of the triangle mesh for the
            % volume preservation code to work. 
            
            nv = mesh.num_vertices;
            v = mesh.getAllVertices();
            he = mesh.getAllHalfedges();
            
            vpos1 = he.from().getTrait('position');
            vpos2 = he.to().getTrait('position');
            vpos3 = he.next().to().getTrait('position');
            
            % this is the part for calculating the volume
            % we have all the vertices and all the half-edges and faces
            
            vol = 1;
        end

This is the formula for calculating the volume:

The mesh is a cell that looks like this:

and each property is the mesh alongside some indexes. For example, vertices are like this:

[Adam Danz]

Even if I flip the z axis in my figure, it still doesn't produce the pyramid coordinates in the image in your comment above.

So, does your previous comment compute the volume of the pyramid would answers your question?

[update Re:PM]

Feel free to attach data to a comment or to your question. But more importantly, please clarify your question. Your code above seems to compute the volume so what problems are you having?

[Hadi Ghahremannezhad]

Yes, because there are also faces and edges in the mesh alongside the vertices.

The code does not compute the volume. The last comment is the part that I need to use the vertex positions to compute the volume of the whole mesh and put it in vol.

Answer 1

[Ridwan Alam]

Not quite sure I understood the problem. Yet, a hunch:

vertices = [-4 0 5;
    1 -5 5;
    1 0 1;
    1 5 5;
    1 0 0];

g = []; N = [];
for k = 1:size(vertices,1)
    g = [g, sum(vertices(k,:))/3];
    a = abs(vertices(k,1)-vertices(k,2));
    b = abs(vertices(k,1)-vertices(k,3));
    N = [N, a*b*sin(pi/3)]; % assuming theta = pi/3
end
Volume = sum(g.*N)/6;

[Hadi Ghahremannezhad]

Do you know what the logical and sign means between two vectors? Is that the normal of two vectors?

[Ridwan Alam]

I believe that meant cross product of two vectors;

A ^ B = |A| |B| sin[theta]

Btw, please vote up if you liked the conversation!