Calculation of volume and centre of mass of an arbitrary polyhedron

Question

Hi I am developing a thesis that will calculate the volume and center of mass of an arbitrary block of rock.

1- The calculation starts with triple volume integrals. The formulas are transformed to line integrals using the divergence theorem and Green theorem. This is a proven method and I have developed software that does the calculations. The input to the software is the coordinates of the vertices of the faces.

2- From the field, the orientations of the planes that define the block of rock are measured using dip and dip direction. These are simply angles that perfectly identify the planes in space. Additionally, the coordinates of a point in the planes are measured. This information allows us to obtain the equation of all the planes that define the block. These planes are, for example, faults, fractures or geologic layers.

3- The combination of the planes in groups of three, without repetition, gives us the coordinates of all vertices and the faces they belong to.

4- The last piece of the puzzle is to figure out the sequence of the coordinates in the face in counterclockwise direction. In other words, it is necessary to create an index of the coordinates so we know in what order they are on the face. Moreover, it is required to determine what vertices really belong to the polyhedron.

I think I can use linear programming to define what vertices (defined by 3 intercepting planes) are really part of the polyhedron. I wont't have an objective function and I don't know how to define the constraints.

Any suggestion? ideas?

Thanks Jair Santos UBC-Canada

Answer 1

Suppose that the planes given are $(a_i, n_i)$ where $a_i$ is a point on the plane and $n_i$ is the outward normal. Then, the region inside the polygon is given by the feasible region of the set of inequalities $\forall i: n_i.x \le n_i.a_i$.

If you do not care about an exact solution (your measurements of the plane would have some errors anyway) I would suggest you go for an approximation algorithm. For example, find a box bounding the polygon. Randomly sample points in the box and estimate the probability of it being inside the polygon. The approximate volume is just the estimated probability times the volume of the box.

Answer 2

This isn't really an answer, but I wasn't sure that I could put a picture in a comment.

I think you need more information. The plane equations by themselves don't properly define the object. For example, the following two objects both have the same 5 infinite planes:

On a more positive note, there are well-established methods (and available code) for computing volumes of polyhedra. For example, see this document and the accompanying software.