The nature and invariant properties of the elliptic polarity of screws in Euclidean space is established using a novel series of mappings, central to which is a quaternion representation. The role of the elliptic polarity in modeling constrained motion is detailed and illustrated by way of an example which has application in the control of robot manipulators. Ball’s planar representation of the two system of screws is generalized and is shown to be a representation on a complex plane where the elliptic polarity is a conformal mapping.

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