  In this work we propose algorithms to explicitly construct a conservative
estimate of the configuration spaces of rigid objects in 2D and 3D. Our
approach is able to detect compact path components and narrow passages in
configuration space which are important for applications in robotic
manipulation and path planning. Moreover, as we demonstrate, they are also
applicable to identification of molecular cages in chemistry. Our algorithms
are based on a decomposition of the resulting 3 and 6 dimensional configuration
spaces into slices corresponding to a finite sample of fixed orientations in
configuration space. We utilize dual diagrams of unions of balls and uniform
grids of orientations to approximate the configuration space. We carry out
experiments to evaluate the computational efficiency on a set of objects with
different geometric features thus demonstrating that our approach is applicable
to different object shapes. We investigate the performance of our algorithm by
computing increasingly fine-grained approximations of the object's
configuration space.
