This paper investigates the inverse and forward kinematic expressions of a three degrees-of-freedom parallel mechanism with identical limb structures, performing three independent rotations. The forward kinematic and constraint expressions are explored in two different spaces, namely, seven-dimensional kinematic space and Euclidean space. The algorithm applied to obtain the forward kinematic and constraint equations in seven-dimensional space, by the means of Study parameters, is a novel approach which is the so-called linear implicitization algorithm. Moreover, the working and assembly modes are computed upon writing the kinematic expression in Euclidean space.