C1 and G1 continuous rational motions using a conformal geometric algebra

Ben Cross, Robert J. Cripps, Glen Mullineux

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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.

Original languageEnglish
Article number114280
JournalJournal of Computational and Applied Mathematics
Early online date29 Mar 2022
Publication statusPublished - 1 Oct 2022


  • Dual quaternions
  • Geometric algebra
  • Geometric continuity
  • Motion design
  • Quaternions
  • Rational motion

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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