Constraint equations of a parallel manipulator can be used to analyze their kinematic behaviour. This paper deals with the determination of the algebraic constraint equations of a 3-RUU parallel manipulator with two approaches. The first one is based on the manipulator geometry and the second one uses the Linear Implicitization Algorithm. The obtained constraint equations through the former approach can be given a geometrical interpretation while the latter approach is less prone to missing physical constraints. Both the ideals of constraint polynomials should lead to the same variety. Furthermore, the simplest set of equations is chosen to solve the direct kinematics problem. For the manipulator under study, it turns out that its direct kinematics problem leads to a factorisable univariate polynomial and a translational operation mode appears.