After discussing the Study point transformation operator, a unified way to formulate kinematic problems, using “points moving on planes or spheres” constraint equations, is introduced. Application to the direct kinematics problem solution of a number of different parallel Schönflies motion robots is then developed. Certain not widely used but useful tools of algebraic geometry are explained and applied for this purpose. These constraints and tools are also applied to some special parallel robots called “double triangular” to show that the approach is flexible and universally pertinent to manipulator kinematics in reducing the complexity of some previously achieved solutions. Finally a novel two-legged Schönflies architecture is revealed to emphasize that good design is not only essential to good performance but also to easily solve kinematic models. In this example architecture, with double basally actuated legs so as to minimize moving mass, the univariate polynomial solution turns out to be simplest, i.e., of degree 2.