This paper studies mappings of spatial kinematics and the geometry for which a spatial displacement is an element. Study’s soma is reviewed and it is shown that Euclidean geometry in three-space with spatial displacements as elements corresponds to elliptic geometry of points in a projective dual three-space. Study’s eight parameters are used to define the mapping of spatial kinematics into points of this projective dual three-space. The basic geometric properties of this dual three-space representation of Study’s soma is developed and it is applied to the study of spatial motions and mechanisms.

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