## Abstract

A combination \(\varvec{a}+{\mathrm{i}}\varvec{b}\) where \({\mathrm{i}}^2=-1\) and \(\varvec{a},\,\varvec{b}\) are real vectors is called a *bivector*. Gibbs developed a theory of bivectors, in which he associated an ellipse with each bivector. He obtained results relating pairs of conjugate semi-diameters and in particular considered the implications of the scalar product of two bivectors being zero. This paper is an attempt to develop a similar formulation for hyperbolas by the use of *jay-vectors*—a jay-vector is a linear combination \(\varvec{a}+{\mathrm{j}}\varvec{b}\) of real vectors \(\varvec{a}\) and \(\varvec{b}\), where \({\mathrm{j}}^2=+1\) but \({\mathrm{j}}\) is not a real number, so \({\mathrm{j}}\ne \pm 1\). The implications of the vanishing of the scalar product of two jay-vectors is also considered. We show how to generate a triple of conjugate semi-diameters of an ellipsoid from any orthonormal triad. We also see how to generate in a similar manner a triple of conjugate semi-diameters of a hyperboloid and its conjugate hyperboloid. The role of complex rotations is discussed briefly. Application is made to second order elliptic and hyperbolic partial differential equations.

## Keywords

Split complex numbers Hyperbolic numbers Coquaternions Conjugate semi-diameters Hyperboloids and ellipsoids PDEs## Notes

## References

- 1.Bell, R.J.T.: Coordinate Geometry of Three Dimensions. Macmillan, London (1937)Google Scholar
- 2.Boulanger, Ph, Hayes, M.: Bivectors and Waves in Mechanics and Optics. Chapman and Hall, London (1993)CrossRefzbMATHGoogle Scholar
- 3.Catoni, F., Boccaletti, D., Cannata, R., Catoni, V., Zampetti, P.: Geometry of Minkowski Space-Time, Chapter 2: Hyperbolic Numbers. Springer, Basel (2011)CrossRefzbMATHGoogle Scholar
- 4.Cockle, J.: On certain functions resembling quaternions and on a new imaginary in algebra. Philos. Mag. Ser. 3
**33**, 435–439 (1848)Google Scholar - 5.Cockle, J.: On systems of algebra involving more than one imaginary; and on equations of the fifth degree. Philos. Mag. Ser. 3
**35**, 434–437 (1849)Google Scholar - 6.Gantmacher, F.R.: Applications of the Theory of Matrices, vol. 2. Interscience Publishers, New York (1959)zbMATHGoogle Scholar
- 7.Gibbs, J. W.: Elements of Vector Analysis, 1881, 1884 (privately printed), OR pp. 17–90, Vol. 2, Part 2 Scientific Papers, Dover Publications, New York (1961)Google Scholar
- 8.Hamilton, W.R.: Lectures on Quaternions. Hodges and Smith, Dublin (1853)Google Scholar
- 9.Hayes, M.: Inhomogeneous plane waves. Arch. Rational Mech. Anal.
**85**, 41–79 (1984)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Hayes, M.: Inhomogeneous electromagnetic plane waves in crystals. Arch. Rational Mech. Anal.
**97**, 221–260 (1987)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Wikipedia
*Split-Complex Number*, https://en.wikipedia.org/wiki/Split-complex number - 12.Yaglom, I.M.: Complex Numbers in Geometry. Academic Press, London (1968)zbMATHGoogle Scholar