Quaternions have long been a powerful tool for describing the rotational motion of rigid bodies including conventional robotic manipulators. In this paper, we demonstrate that the quaternionic representation of kinematics for rigid link manipulators can be applied to continuum manipulators by passing to the limit as the link lengths tend to zero. Compact expressions are presented for forward kinematics that are based on the notions of distributed rotation quaternion and relative curvature. The robot’s pose in this formulation can be described by an arbitrary piecewise smooth function. Some simple closedform examples are discussed, and an approach to solving the inverse kinematics problem is outlined based on a modification of the cyclic coordinate descent algorithm. This work has been submitted to the IEEE for possible publication.  Copyright may be transferred without notice, after which this version  may no longer be accessible.