Get the surface area of a polyhedron (3D object)

I have a 3D surface, (think about the xy plane). The plane can be slanted. (think about a slope road).

Given a list of 3D coordinates that define the surface(Point3D1X, Point3D1Y, Point3D1Z, Point3D12X, Point3D2Y, Point3D2Z, Point3D3X, Point3D3Y, Point3D3Z, and so on), how to calculate the area of the surface?

Note that my question here is analogous to finding area in 2D plane. In 2D plane we have a list of points that defines a polygon, and using this list of points we can find the area of the polygon. Now assuming that all these points have z values in such a way that they are elevated in 3D to form a surface. My question is how to find the area of that 3D surface?

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I don't know about optimising this method (I've not done it in code before), but the way to approach it mathematically is to split your shape into triangles, the area of which is then easily calculated and summed. (Remember: area of a triangle is width * height * 0.5 - you'll need to calculate the height of non-right-angled triangles.)

Doing these things in 3D generally means one more calculation is needed at each stage. For example, in 2D, the distance between 2 points (the length of a side of your shape) is calculated something like this (pseduocode because I don't have VS on this machine):

double DistanceBetween(Point a, Point b)
{
   double dx = a.x - b.x;
   double dy = a.y - b.y;
   return SquareRoot(dx*dx + dy*dy);
}

In three dimensions that becomes:

double DistanceBetween(Point3d a, Point3d b)
{
   double dx = a.x - b.x;
   double dy = a.y - b.y;
   double dz = a.z - b.z;
   return SquareRoot(dx*dx + dy*dy + dz*dz);
}

Splitting a shape into arbitrary triangles just involves picking any three adjacent vertices at a time, until your're down to your last three.

answered 2010-02-28T09:35:23.680

Get the surface area of a polyhedron (3D object)

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