How to calculate the volume enclosed by a set of XYZ points in 3D?

Hi,

I am trying to find the volume of a region (defined by X,Y and Z coordinates) enclosed below a (Z='constant') plane. The data has peaks (positive Z) and a valley (negative Z), and the mean surface is assigned z=0. I have tried with the following piece of code, but I doubt it gives me the total volume bound by the surface against z=0, including the peak volumes as well.

            [X,Y,Z] = xyzread("data.xyz");
            plot3(X,Y,Z)
         
            [fitobject, gof, output] = fit([X,Y],Z, 'biharmonicinterp');
            plot(fitobject)
            
            a = min(X);
            b = max(X);
            c = min(Y);
            d = max(Y);
            
            volume_under_fit = quad2d(fitobject,a,b,c,d)

Answer 1

[Bruno Luong]

V is the volume between the plane x-y (z==0) and the surface z(x,y) from your data.

If you want the volume of the data after substract the base plane surface, you need to estimated the equation by regression.

load('data.xyz')

x=data(:,1);
y=data(:,2);
z=data(:,3);
T=delaunay(x,y);
trisurf(T,x,y,z);

xy = [x,y];
a = xy(T(:,2),:)-xy(T(:,1),:);
b = xy(T(:,3),:)-xy(T(:,1),:);
V = ((a(:,1).*b(:,2)-a(:,2).*b(:,1))' * sum(z(T),2))/6

[Bruno Luong]

"V is the volume between the plane x-y (z==0) and the surface z(x,y) from your data."

V is the (signed) integral of the whole thing.

If you filter out the positive surface, you get smaller volume.

The formula computed the integral, suming up the positive and negative. Up to you to separate negative and positive volumes.

If you have a question of YOUR result, show us what you have done with the code.

[Bruno Luong]

load('data.xyz')
x=data(:,1);
y=data(:,2);
z=data(:,3);
T=delaunay(x,y);
xy=[x,y];
a = xy(T(:,2),:)- xy(T(:,1),:);
b = xy(T(:,3),:)- xy(T(:,1),:);
A = (a(:,1).*b(:,2)-a(:,2).*b(:,1))/2;
h =  mean(z(T),2);
pos = h > 0;
sh = trisurf(T,x,y,z,'CData',double(pos.'),'EdgeColor','none');
colormap(jet)

Vt = A .* h;
Vpos = sum(Vt(pos))     % sign (+) volume of of the red part
Vneg = sum(Vt(~pos))    % sign (-) volume of of the blue part
V = Vpos+Vneg      

[Sourav Sahoo]

Hi,

Thanks a lot for your help! It works!!

[Lyhour Chhay]

Dear Bruno Luong and Sourav Sahoo,

First of all, I really interest your approach to calculate the volme. I have a problem similar to a proposed problem. I have point cloud with x,y,z data. I want to find the volume between the plan x-y (z==5) and my point cloud data. I will show in the figure below. how can I solve this problem ? I tried to use the code in the comment, it show the negative value for my result. thank you very much.

[Bruno Luong]

If you want to adapt the above code plane z=5 and you data is bellow the plane then change to

x=data(:,1);
y=data(:,2);
z=5-data(:,3);

and do the rest similarly.

Answer 2

[Walter Roberson]

https://www.mathworks.com/support/matlabcentral/answers/489219-alternatives-to-delaunay-triangulation