



 
For the moment, let's suppose the object to be convex and to contain the origin. Its volume is the sum of the volumes of all tetrahedra bounded by the origin and a mesh triangle. These can be found using determinants; the volume of the tetrahedron with vertices (0,0,0), (a_{1},a_{2},a_{3}), (b_{1},b_{2},b_{3}) and (c_{1},c_{2},c_{3}) is [a_{1}b_{2}c_{3} +a_{2}b_{3}c_{1} + a_{3}b_{1}c_{2} a_{1}b_{3}c_{2} a_{2}b_{1}c_{3}  a_{3}b_{2}c_{1}]/6. Make sure to get the orders consistent (eg, each triangle's vertices enumerated in clockwise order as seen from outside) or the volume contributed may be negative in some cases. If the object is not convex or does not contain the origin, it turns out that this makes no difference. If the origin "sees" the mesh triangle from outside the contribution is negative & the "cone" joining the object to the origin is thus cancelled out. Good Hunting! Ozen wrote back
Ozen, "The 3D closed mesh object is a car model. That is, the 3D model has convexity and concavity."
"In addition to its volume, it is also required the center of volume."
Good hunting!  


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. 