Visualizing quaternion rotation

ABSTRACT

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.

REFERENCES

• 1: Alan H. Barr , Bena Currin , Steven Gabriel , John F. Hughes, Smooth interpolation of orientations with angular velocity constraints using quaternions, ACM SIGGRAPH Computer Graphics, v.26 n.2, p.313-320, July 1992
• 2: Alan H. Barr, Global and local deformations of solid primitives, ACM SIGGRAPH Computer Graphics, v.18 n.3, p.21-30, July 1984
• 3: B()I~K}:R, E. 1973. The spinor spanner. Am. Math. Month. 80, 9 (Nov.), 977 984.
• 4: CAYIJ~;Y, A. 1897. The Collected Mathematical Papers af Arthur Cayley. Cambridge University Press, Cambridge, U.K.
• 5: FItnN('Is, G.K. 1987. A Topological Picturebook. Springer-Verlag, New York.
• 6: FICAN4'IS, G. K. ANI) KAUFFMAN, L.H. 1994. Air on the Dirae strings. In Mathematical Legacy of Wilhelm Magnus. AMS, Providence, R.I.
• 7: HAMII,'I'{)N, W.R. 1866. Elements af Quaternions. Longmans Green, Lond()n.
• 8: Andrew J. Hanson, The rolling ball, Graphics Gems III, Academic Press Professional, Inc., San Diego, CA, 1992
• 9: HART, J. C. 1993. Sphere tracing: Simple robust antialiased rendering of distance-based implicit surfaces. Tech. Pep. EECS-93-15, School of EECS, Washington State Univ., Pu}lman, Wash. Appears in SIGGRAPH '93, Course Notes #25 "Modeling, Visualizing and Animating Implicit Surfaces."
• 10: D. Kalra , A. H. Barr, Guaranteed ray intersections with implicit surfaces, ACM SIGGRAPH Computer Graphics, v.23 n.3, p.297-306, July 1989
• 11: KAUF~'MAN, L.H. 1991. Knots and Physics. World Scientific, Teaneck, N.J.
• 12: KAUrFMAN, L.H. 1987. On Knots. Princeton University Press, Princeton, N.J.
• 13: MISNER, C. Wl, THORNE, K. S., AND WHEELER, J.A. 1973. Grat, itation. Freeman, San Francisco, Calif.
• 14: SANDm, D. J., KAUFFMAN, L. H., AND FRAr~CIS, G.K. 1993. Air on the Dirac strings. SIGGRAPH Video Rev. 93. Animation.
• 15: Ken Shoemake, ARCBALL: a user interface for specifying three-dimensional orientation using a mouse, Proceedings of the conference on Graphics interface '92, p.151-156, September 1992, Vancouver, British Columbia, Canada
• 16: Ken Shoemake, Animating rotation with quaternion curves, ACM SIGGRAPH Computer Graphics, v.19 n.3, p.245-254, Jul. 1985
• 17: SPIVAK, M. 1965. Calculus on Manifolds. W. A. Benjamin, New York.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.5 Computational Geometry and Object Modeling
          Subjects: Hierarchy and geometric transformations

Additional Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.6 Methodology and Techniques
          Subjects: Graphics data structures and data types

General Terms:
Algorithms, Theory

Keywords:
deformation, education, interpolation, orientation, quaternions, rotation, visualization