Visualizing Quaternion Rotation

John C. Hart, George K. Francis, Louis H. Kauffman

Research output: Contribution to journalArticlepeer-review


Quaternions play a vital role in the representation of rotations in computer graphics, primarily for animation and user interfaces. Unfortunately, quaternion rotation is often left as an advanced topic in computer graphics education due to difficulties in portraying the four-dimensional space of the quaternions. One tool for overcoming these obstacles is the quaternion demonstrator, a physical visual aid consisting primarily of a belt. Every quaternion used to specify a rotation can be represented by fixing one end of the belt and rotating the other. Multiplication of quaternions is demonstrated by the composition of rotations, and the resulting twists in the belt depict visually how quaternions interpolate rotation. This article introduces to computer graphics the exponential notation that mathematicians have used to represent unit quaternions. Exponential notation combines the angle and axis of the rotation into concise quaternion expression. This notation allows the article to present more clearly a mechanical quaternion demonstrator consisting of a ribbon and a tag, and develop a computer simulation suitable for interactive educational packages. Local deformations and the belt trick are used to minimize the ribbon's twisting and simulate a natural-appearing interactive quaternion demonstrator.

Original languageEnglish (US)
Pages (from-to)256-276
Number of pages21
JournalACM Transactions on Graphics (TOG)
Issue number3
StatePublished - Jan 7 1994


  • deformation
  • education
  • interpolation
  • orientation
  • quaternions
  • rotation
  • visualization

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design


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