John C. Hart
George K. Francis
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.
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Index Terms
- Visualizing quaternion rotation
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Reviews
Nickolas S. Sapidis
The authors discuss the fundamentals of quaternions, their use in computer graphics for specifying rotations, and the mathematical modeling of a quaternion demonstrator. Section 2 begins with a short review of definitions and other mathematical results on quaternions, then proceeds to apply them to rotation representation, and finally focuses on the quaternion demonstrator, a mechanical unit quaternion multiplier. This consists of a ribbon (belt), with one end fixed and the other end free, and a tag fastened to the free end of the belt. The orientations of the tag, along with the twists in the belt, represent the unit quaternions. The modeling of various quaternion operations (including negation and multiplication) is discussed in detail. Section 3 focuses on the belt trick and its topological simulation, which is based on a globally specified deformation. Section 4 is this paper's original contribution, where techniques from differential geometry are used for modeling the quaternion demonstrator, and for regulating the twists and motions of the belt. More specifically, the authors represent the belt as a family of rotating line segments whose orientations interpolate the orientation of the belt's fixed end into the orientation of the tag. This idea is further developed, finally producing an analytic expression for the spine of the belt. The last part of this section focuses on issues related to implementing the above model, of which the most important is regulating the rate at which the quaternion demonstrator performs automatic belt tricks. For this purpose, the authors define a unit quaternion in spherical coordinates and derive a formula relating the incremental rotation of the tag to the maximum allowed rate of change of the belt. Further details on the authors' implementation are given in section 5, while their rendering procedure is described in the appendix. This is an important contribution, as the quaternion is not yet a standard tool in computer graphics. It seems very useful for representing rotations of objects and probably for solving other problems.
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