- A new family of orientation parameters derived from the Euler parameters is presented. They are found by a general stereographic projection of the Euler parameter constraint surface, a four-dimensional unit sphere, onto a three-dimensional hyperplane. The resulting set of three stereographic parameters have a low degree polynomial nonlinearity in the corresponding kinematic equations and direction cosine matrix parameterization. The stereographic parameters are not unique, but have a corresponding set of "shadow" parameters. These shadow parameters are distinct, yet represent the same physical orientation. Using the original stereographic parameters combined with judicious switching to their shadow set, it is possible to describe any rotation without encountering a singularity. The symmetric stereographic parameters are nonsingular for up to a principal rotation of 360. The asymmetric stereographic parameters are well suited for describing the kinematics of spinning bodies, since they only go singular when oriented at a specific angle about a specific axis. A globally regular and stable control law using symmetric stereographic parameters is presented which can bring a spinning body to rest in any desired orientation without backtracking the motion.