This paper looks at the forward kinematics problem of the 3-RPS manipulator from a geometric perspective. It shows that the problem is equivalent to finding the intersection of a pair of quad-circular octic curves with a circle. The results explain all the known algebraic results in this regard, and provide an intuitive insight into the nature of the solutions, as regards the operation modes, and the assembly modes inside each. The theoretical results are illustrated with a numerical example, where all the 16 assembly modes are real. © Springer International Publishing AG 2018.