Comparison of the finite helical axis and the rectangular coordinate system in representing orthodontic tooth movement

Kazuo Hayashi, Ralph DeLong, Itaru Mizoguchi

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


In orthodontics, tooth movement is typically described using the rectangular coordinate system (XYZ); however, this system has several disadvantages when performing biomechanical analyses. An alternative method is the finite helical axis (FHA) system, which describes movement as a rotation about and a translation along a single axis located in space. The purpose of this study was to examine differences between the FHA and the XYZ systems in analyzing orthodontic tooth movement. Maxillary canine retraction was done using sliding mechanics or a retraction spring with midpalatal orthodontic implants used as measuring references. Tooth movement calculated with the FHA was compared with the corresponding movement in the rectangular coordinate system weekly over a 2-month interval in eight patients. The FHA showed that sliding mechanics controlled rotation of the canine better than the retraction spring (Ricketts retractor), and that the Ricketts retractor controlled tipping better. Changes in the FHA direction and position vectors with time showed that the biomechanical forces are not uniform during the treatment period. In both mechanics, the FHA provided a simple biomechanical model for canine retraction.

Original languageEnglish (US)
Pages (from-to)2925-2933
Number of pages9
JournalJournal of Biomechanics
Issue number16
StatePublished - 2006

Bibliographical note

Funding Information:
This work was supported by the standard financial plan of the Health Sciences University of Hokkaido, and the Minnesota Dental Research Center for Biomaterials and Biomechanics, University of Minnesota School of Dentistry.


  • Finite helical axis
  • Ricketts
  • Sliding mechanics
  • Tooth movement


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