  We describe and analyze some Monte Carlo methods for manifolds in Euclidean
space defined by equality and inequality constraints. First, we give an MCMC
sampler for probability distributions defined by un-normalized densities on
such manifolds. The sampler uses a specific orthogonal projection to the
surface that requires only information about the tangent space to the manifold,
obtainable from first derivatives of the constraint functions, hence avoiding
the need for curvature information or second derivatives. Second, we use the
sampler to develop a multi-stage algorithm to compute integrals over such
manifolds. We provide single-run error estimates that avoid the need for
multiple independent runs. Computational experiments on various test problems
show that the algorithms and error estimates work in practice. The method is
applied to compute the entropies of different sticky hard sphere systems. These
predict the temperature or interaction energy at which loops of hard sticky
spheres become preferable to chains.
