Compute area of 3D surface given coordinates of points

Question

Given the coordinates of each point, how to calculate the area of a three-dimensional surface in MATLAB?

x=-5:1:5;y=-5:1:5;
[xx,yy]=meshgrid(x,y);zz=Z;
figure(1)
mesh(xx,yy,zz) 
figure(2)
xb=-5:0.25:5;
yb=-5:0.25:5;
[xxb,yyb]=meshgrid(xb,yb);
zzb=interp2(xx,yy,zz,xxb,yyb,'cubic');
mesh(xxb,yyb,zzb)

above is my code.

the picture is the data of z.

I tried to search for questions alike but I just couldn't find it.

Answer 1

1.- alphaShape does not work for this question because this particular surface is open.

alphaShape generates 2D or 3D closed POLYGONS.

This is the 1st surface

x=-5:1:5;y=-5:1:5;
[xx,yy]=meshgrid(x,y);

zz=[13.6 -8.2 -14.8 -6.6 1.4 0 -3.8 1.4 13.6 16.8 0 
-8.2 -15.8 -7.9 2.2 3.8 0 0.6 7.3 10.1 0 -16.8 
-14.8 -7.9 2.5 5.8 2.3 0 2.7 5.1 0 -10.1 -13.7 
-6.6 2.2 5.9 3.0 -0.3 0 1.9 0 -5.1 -7.3 -1.4 
1.4 3.8 2.3 -0.3 -0.9 0 0 -1.7 -2.7 -0.6 3.8 
0 0 0 0 0 0 0 0 0 0 0 
-3.8 0.6 2.7 1.7 0 0 0.9 0.3 -2.3 -3.8 -1.4 
1.4 7.3 5.1 0 -1.7 0 0.3 -3.1 -5.8 -2.2 6.6 
13.6 10.1 0 -5.1 -2.7 0 -2.3 -5.8 -2.5 7.9 14.8 
16.8 0 -10.1 -7.3 -0.6 0 -3.8 -2.2 7.9 15.8 8.2 
0 16.3 -13.6 -1.4 3.8 0 -1.4 6.6 14.8 8.2 -13.6];

figure(1)
sh1=mesh(xx,yy,zz) 

And this is the 2nd surface

figure(2)
xb=-5:0.1:5;
yb=-5:0.1:5;
[xxb,yyb]=meshgrid(xb,yb);
zzb=interp2(xx,yy,zz,xxb,yyb,'cubic');
mesh(xxb,yyb,zzb)

Now try alphaShape on the 1st surface

P1=[xx(:) yy(:) zz(:)];
shp1 = alphaShape(P1(:,1),P1(:,2),P1(:,3));
figure;
plot(shp1)
axis equal

voila : here is the contraption

and when attempting

A1=area(shp1)

bang :

Error using alphaShape/area
This method is only applicable to 2D alpha shapes.

2.- One has to integrate adding up each flat surface.

The vertexs of each flat surface are available using the handle available from mesh

sh1=mesh(xx,yy,zz) 

check the abridged properties

sh1 = 
  Surface with properties:

   EdgeColor: 'flat'
   LineStyle: '-'
   FaceColor: [1 1 1]
FaceLighting: 'none'
   FaceAlpha: 1
       XData: [11×11 double]
       YData: [11×11 double]
       ZData: [11×11 double]
       CData: [11×11 double]

click on Show all properties that shows up right after the abridged properties in Command Window to read all properties

          AlignVertexCenters: off
                   AlphaData: 1
            AlphaDataMapping: 'scaled'
             AmbientStrength: 0.300000000000000
                  Annotation: [1×1 matlab.graphics.eventdata.Annotation]
            BackFaceLighting: 'reverselit'
                BeingDeleted: off
                  BusyAction: 'queue'
               ButtonDownFcn: ''
                       CData: [11×11 double]
                CDataMapping: 'scaled'
                   CDataMode: 'auto'
...

           VertexNormals: [11×11×3 double]
       VertexNormalsMode: 'auto'
                 Visible: on
                   XData: [11×11 double]
               XDataMode: 'manual'
             XDataSource: ''
                   YData: [11×11 double]
               YDataMode: 'manual'
             YDataSource: ''
                   ZData: [11×11 double]
             ZDataSource: ''

3.- Good News : In The Mathworks File Exchange there is a function called surfarea, thanks to Sky Santorius, that does exactly the task mentioned in point 2 which is what 石sh asks for.

surfarea is available in the following link :

https://uk.mathworks.com/matlabcentral/fileexchange/62992-surface-area?s_tid=srchtitle_surface%20area_2

Applying surfare to 1st surface

[A1,cell_areas_A1,centroid1]=surfarea(xx,yy,zz);

the resulting surface area is

A1 =
     6.783763291711880e+02

and when calculating area of the interpolated 2nd surface

[A2,cell_areas_A2,centroid2]=surfarea(xxb,yyb,zzb);

the resulting surface area is

A2 =
     6.504878453848488e+02

Answer 2

Try converting it to an AlphaShape then use SurfaceArea.