On the geometric interpretation of quaternions by cones

Georgiy Scheglov

doi:doi:10.12737/2308-4898-2022-10-3-23-34

UDC 514

On the geometric interpretation of quaternions by cones

Published в Geometry & Graphics · Volume 10, Issue 3, 2022 · Pages 23–34 · Rubrics: Scientific problems of geometry

DOI 10.12737/2308-4898-2022-10-3-23-34

Received: 20.09.2022 Accepted: 26.09.2022 Published: 26.09.2022

Authors

Georgiy Scheglov ¹

1 Bauman Moscow State Technical University (Professor)
Moscow, Russian Federation

ABSTRACT

The geometric interpretation of quaternions is considered. The visualization complexity of quaternions is due to the fact that these objects have four independent parameters. A literature analysis shows that the problem of geometric interpretation of quaternions has not been completely solved to date. The first section provides general provisions on quaternions and the necessary notations. The second section describes the classical geometric interpretation of quaternions by arcs on a unit sphere. The third section describes a new geometric interpretation and its application to the problem of a vector finite rotation. The geometric interpretation of the quaternion as the surface of a right circular cone is presented. This representation allow demonstrating it as a holistic object in which the scalar and vector parts are interconnected, taking into account their modules and signs. For the considered normalized quaternion, it is easy to understanding an important entity, the quaternion versor: in general, it is a cone, which in the limiting case of a pure scalar quaternion transform into a sphere, and in the limiting case of a pure vector quaternion transform into an ordinary vector. This distinctive feature of the proposed geometric interpretation makes it possible, even when projected onto a plane, to clearly distinguish visualization of the quaternions with a nonzero scalar part from pure vector quaternions, which is difficult to do in the other known interpretations. The representation of quaternions by cones clearly demonstrates the need for a double quaternion product, when the vector is rotated around an arbitrary axis. Images of quaternions as cones, spheres and vectors can be useful in the study of quaternion algebra, which is currently finding increasing use in engineering.

KEYWORDS

geometry quaternions geometric interpretation product of quaternions education

Information References

Authors:

1. Bauman Moscow State Technical University (Professor)

Moscow, Russian Federation

Type:

Article

Pages:

from 23 to 34

Status:

Published

Subject area:

UDC 514

Language:

Russian

Keywords:

geometry, quaternions, geometric interpretation, product of quaternions, education

References

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