Gradient angle estimation by uniform directional simulation on a cone

A sample of uniformly distributed unit vectors on an n-dimensional spherical cone is generated. The distances to a given limit state surface in the directions of the unit vectors of the sample are calculated and each of these distances are projected on the cone axis. The theoretical distribution of these projections is derived assuming the limit-state surface to be a hyperplane. This distribution depends on the angle between the cone axis and the normal vector to the hyperplane. Assuming sufficient flatness of the actual limit-state surface within a neighbourhood of the cut point with the cone axis, the cone top angle can be chosen small enough that this distribution can be taken as the basis for the formulation of the likelihood function of the angle given the sample of projections. The angle of maximum likelihood is then the indicator of whether the cut point can be taken as a sufficiently accurate approximation to a locally most central limit state point. Moreover, the estimated angle can be used to correct the geometric reliability index.