Efficient Computation of Body Moments

Institute of Engineering Cybernetics Academy of Sciences of Republic Belarus Surganova 6, 220012 Minsk, Belarus {tuzikov,sheynin,vasiliev}@mpen.bas-net.by Received April 10, 2002 It is well known that such important geometric characteristics of 3D bodies as volume and orientation can be defined in terms of moments. A moment computation depends greatly on a body representation. The most popular representation is a polygonal representation. In this case an object is represented by a mesh of polygonal facets. The report presents explicit formulae for calculation of volume and surface moments for 3D polyhedral shapes. The formulae can be generalized for polytopes in Rn. We discuss also efficient algorithms for calculation of 3D body volume and surface moments. The algorithms are based on the proposed formulae and take advantages of a polygonal representation. They use only coordinates of body vertices and faces orientation. The way to compute a moment of a polyhedral shape is to compute first the moment of a tetrahedron with one vertex in the origin. We derive the formula for volume moments of arbitrary order using the Dirichlet integral and a linear substitution transforming an arbitrary tetrahedron to the coordinate tetrahedron. To compute a surface moment on a tetrahedron face we also use the Gauss-Ostrogradsky formula. A moment of an arbitrary polyhedron is computed as the sum of oriented moments of tetrahedra with one vertex in the coordinate origin and the opposite face constructed by a triangulation of every face of a polyhedron. This procedure and the resulting formula does not depend on the mutual position of vertices and faces of a polyhedron. Let P is a 3D polyhedron with n faces, and face i has si vertices (xi,0, yi,0, zi,0), . . . , (xi,si−1, yi,si−1, zi,si−1) numbered in a counter-clockwise order with respect to the outer normal. Then the following formula is true for a volume moment mpqrV of order p + q + r [1]: