Uniform distribution of points on hypersurfaces: simulation of random equiprobable rotations

The paper describes a universal method for simulation of uniform distributions of points on smooth regular surfaces in Euclidean spaces of various dimensions. The authors give an interpretation of a set of possible values of Rodrigues–Hamilton parameters used to describe a rigid rotation as a set of points of a three-dimensional hypersphere in four-dimensional Euclidean space. The relationship between random equiprobable rotations of a rigid body and a uniform distribution of points on the surface of a three-dimensional hypersphere in four-dimensional Euclidean space is established.