Transformations of Complex, Dual, and Hyperbolic Numbers

type

complex

dual

hyperbolic

This Demonstration compares the multiplicative transformation properties on the

x,y

plane of complex, dual, and hyperbolic numbers. These number types share the form

z=x+βy

, where

x

and

y

are real numbers and

β

has the defining characteristics:

2

β

=-1

for complex numbers,

2

β

=0

for dual numbers, and

2

β

=+1

for hyperbolic numbers. Their conjugates share the form

*

z

=x-βy

and their modulus the form

*

z

z.

The figure shows, for the selected number type,

1. a transform point on its unit-modulus locus (in blue),

2. an object point in the plane (in red), and

3. the point resulting from their multiplicative transform on its trajectory (in brown).

In the complex plane, multiplying by a unit-modulus transform produces a rotation about the origin; in the dual plane, it produces a translation in the pure dual direction, and in the hyperbolic plane, a traversal of an hyperbola that passes through the object point.