General Two-Dimensional Hypercomplex Numbers

Abstract

In this chapter we study the Euclidean and pseudo-Euclidean geometries associated with the general two-dimensional hypercomplex variable, i.e., the algebraic ring (see Section 2.2)

$$ \{ z = x + uy; u^2 = \alpha + u\beta ; x,y,\alpha ,\beta \in R; u \notin R\} , $$

(6.0.1)

and we show that in geometries generated by these numbers, ellipses and general hyperbolas play the role which circles and equilateral hyperbolas play in Euclidean and in pseudo-Euclidean planes, respectively.