Modern manufacturing processes require unambiguous description of product morphology. In spite of numerous successes in the development of new mathematical tools, there not exists a method providing complete and coherent information on the product shape along its lifecycle. Consequently, industrial methods currently employed in dimensional and geometrical controls do not fully satisfy designers, manufactures and customers. A possible solution could be a statistical description of product shape because it has strong mathematical basis, uses powerful analysis tools and provides a single unifying model along the product development process. The comparison with industrial practice and deterministic mathematical tools in the design, manufacturing and verification phases, shows some interesting advantages of the probabilistic approach. The paper illustrates the theoretical basis of the probabilistic approach, provides the instruments necessary to its implementation and, finally, shows some applications in the inspection of mechanical objects.

1.
Srinivasan
,
V.
,
1999
, “
A GPS Language Based on a Classification of Symmetry Groups
,”
Comput.-Aided Des.
,
31
(
11
), pp.
659
668
.
2.
Requicha
,
A.
,
1983
, “
Toward a Theory of Geometric Tolerancing
,”
Int. J. Robot. Res.
,
2
(
2
), pp.
45
60
, Winter.
3.
Van Houten F, Kals, H., Editors, 1999, Global Consistency of Tolerancing, Dordrecht, Kluwer Academic.
4.
Srinivasan, V., 2001, “An Integrated View of Geometrical Product Specification and Verification,” Proc. of the 7th CIRP Int. Seminar on CAT, Cachan, pp. 7–16.
5.
Parzen
,
E.
,
1962
, “
On Estimation of a Probability Density Function and Mode
,”
Expert Sys. Applic.
,
33
, pp.
1065
1076
.
6.
O’Connor, M., Srinivasan, V., 1996, “Connected Lie and Symmetry Subgroups of the Rigid Motions: Foundations and Classification,” IBM Research Report, RC 20512.
7.
Humienny, Z., et al., 2001, Geometrical Product Specification, Warsaw University of Technology Printing House.
8.
Herve´, J., 1976, “La ge´ometrie du groupe des de´placements applique´e a l’analyze cine´matique des me´canismes,” These de Doctorat d’Etat en Sciences Physiques.
9.
Cle´ment, A., Riviere, A., and Temmerman, M., 1994, “Cotation tridimensionnelle des systemes me´caniques,” PYC Edition.
10.
Karger, A., and Novak, J., 1985, Space Kinematics and Lie Groups, Gordon and Breach Science Publishers, New York.
11.
Fukunaga, K., 1972, Introduction to Statistical Pattern Recognition. New York, Academic Press.
12.
El Maragy, H. A., Editor, 1998, Geometric Design Tolerancing: Theories, Standards and Applications, London, Chapman & Hall.