How do I determine the surface area of a 2-D surface in a 3-D space?

I have 3 vectors "xcoord", "ycoord", and "zcoord" that represent the (x,y,z)-coordinates of a 2D surface in 3D space. I want to determine the surface area of the surface.

I tried using the "surfaceArea" method for an "AlphaShape" object, but the surface was disconnected. On changing the "Alpha" value, the 2D surface became a 3D object.

I want to compute the surface area for the connected 2D surface. How can I do this? 

Answer 1

[MathWorks Support Team]

If  you want to compute the surface area of a 2-D surface in a 3-D space, the Delaunay Triangulation would be the best approach to go ahead with. You could compute the sum of the triangles formed by the Delaunay Triangulation to find the surface area of the 2-D surface.

 The following steps should help to obtain a 'delaunay' surface and to compute the surface area of the same.

 1) tri = delaunay(X,Y) creates a 2-D Delaunay triangulation. 'tri' is a matrix representing the set of triangles that make up the triangulation.

tri = delaunay(xcoord,zcoord);

P = [xcoord,ycoord,zcoord];

2) Obtain the edges in each triangle formed by the 'delaunaytriangulation'

v1 = P(tri(:,2), :) - P(tri(:,1), :);
v2 = P(tri(:,3), :) - P(tri(:,2), :);

3) Calculating the cross product of the edges in each triangle of the surface

cp = 0.5*cross(v1,v2);

4) Surface area of the entire surface is calculated as the sum of the areas of the individual triangles

surfaceArea = sum(sqrt(dot(cp, cp, 2)))