General formula for calculating the volume of a polyhedron

Question

Given the list of vertices ( v) and the list of edges connecting the vertices ( e) and the list of surfaces connecting the edges ( s), how to calculate the volume of the Polyhedron?

Answer 1

1. Take the polygons and divide them into triangles.
2. Consider the tetrahedron formed by each triangle and an arbitrary point (origin).
3. Sum the signed volumes of these tetrahedrons.

Notes:

1. This will only work if you can maintain a consistent CW or CCW order for triangles when viewed from the side.
2. The volume with a tetrahedron sign is 1/6 of the determinant of the following matrix:

[x1 x2 x3 x4]
[y1 y2 y3 y4]
[z1 z2 z3 z4]
[1 1 1 1]

- (x, y, z, 1).

, , , , .

, 1/6 .

Answer 2

, , . , .

( , /, , .
http://mathoverflow.net)

Answer 3

-, , .

, "" ( ). , . {h1, h2, h3} - , A - , A (h1 + h2 + h3)/3. , .

, , . , .

, (1) (2) "" "". , , , , . , . xy . , , .

, ? , , , , . , . , -1 ( , , ).

: ! , , , . , , .

Answer 4

, , , . , . TetGen, . , . TetGen , CGAL , . CGAL , .

Answer 5

Here's a potential implementation for Python. Can anyone check if this is correct? I believe that I don’t have enough permutations of the points, because my second test (cube) gives 0.666, not 1. Ideas anyone?

Greetings EL

class Simplex(object):
    '''
    Simplex
    '''


    def __init__(self,coordinates):
        '''
        Constructor
        '''

        if not len(coordinates) == 4:
            raise RuntimeError('You must provide only 4 coordinates!')

        self.coordinates = coordinates



    def volume(self):
        '''
        volume: Return volume of simplex. Formula from http://de.wikipedia.org/wiki/Tetraeder
        '''
        import numpy

        vA = numpy.array(self.coordinates[1]) - numpy.array(self.coordinates[0])
        vB = numpy.array(self.coordinates[2]) - numpy.array(self.coordinates[0])
        vC = numpy.array(self.coordinates[3]) - numpy.array(self.coordinates[0])

        return numpy.abs(numpy.dot(numpy.cross(vA,vB),vC)) / 6.0  


class Polyeder(object):

    def __init__(self,coordinates):
        '''
        Constructor
        '''

        if len(coordinates) < 4:
            raise RuntimeError('You must provide at least 4 coordinates!')

        self.coordinates = coordinates


    def volume(self):

        pivotCoordinate = self.coordinates[0]

        volumeSum = 0

        for i in xrange(1,len(self.coordinates)-3):

            newCoordinates = [pivotCoordinate]

            for j in xrange(i,i+3):
                newCoordinates.append(self.coordinates[j])

            simplex = Simplex(newCoordinates)
            volumeSum += simplex.volume()

        return volumeSum


coords = []

coords.append([0,0,0])
coords.append([1,0,0])
coords.append([0,1,0])
coords.append([0,0,1])

s = Simplex(coords)
print s.volume()

coords.append([0,1,1])
coords.append([1,0,1])
coords.append([1,1,0])
coords.append([1,1,1])

p = Polyeder(coords)
print p.volume()