The main objective of this paper is to study the Euclidean displacement of a 5-DOF parallel mechanism performing three translation and two independent rotations with identical limb structures-recently revealed by performing the type synthesis-in a higher dimensional projective space, rather than relying on classical recipes, such as Cartesian coordinates and Euler angles. In this paper, Study's kinematic mapping is considered which maps the displacements of three-dimensional Euclidean space to points on a quadric, called Study quadric, in a seven-dimensional projective space, P7. The main focus of this contribution is to lay down the essential features of algebraic geometry for our kinematics purposes, where, as case study, a 5-DOF parallel mechanism with identical limb structures is considered. The forward kinematic problem is reviewed and the kinematic mapping is introduced for both general and first-order kinematics, i.e., velocity, which provides some insight into the better understanding of the kinematic behaviour of the mechanisms under study in some particular configurations for the rotation of the platform and also the constant-position workspace.

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