C¹ and G¹ continuous rational motions using a conformal geometric algebra

Traditional rational motion design describes separately the translation of a reference point in a body and the rotation of the body about it. This means that there is dependence upon the choice of reference point. When considering the derivative of a motion, some approaches require the transform to be unitary. This paper resolves these issues by establishing means for constructing free-form motions from specified control poses using multiplicative and additive approaches. It also establishes the derivative of a motion in the more general non-unitary case. This leads to a characterization of the motion at the end of a motion segment in terms of the end pose and the linear and angular velocity and this, in turn, leads to the ability to join motion segments together with either C¹- or G¹-continuity.