Motor Variable - Elementary Functions of A Motor Variable

Elementary Functions of A Motor Variable

Let D =, the split-complex plane. The following exemplar functions f have domain and range in D:

The action of a hyperbolic versor is combined with translation to produce the affine transformation

.When c = 0, the function is equivalent to a squeeze mapping.

The squaring function has no analogy in ordinary complex arithmetic. Let

and note that

The result is that the four quadrants are mapped into one, the identity component:

.

Note that forms the unit hyperbola . Thus the reciprocation

involves the hyperbola as curve of reference as opposed to the circle in C.

On the extended complex plane one has the class of functions called Mobius transformations. For the motor variable the analogous functions are constructed with inversive ring geometry:

The extension of the split-complex plane to make these mappings bijective is a part of the subject of inversive split-complex geometry.

On the ordinary complex plane, the Cayley transform carries the upper half-plane to the unit disk, thus bounding it. A mapping of the identity component U₁ into a rectangle provides a comparable bounding action:

where T = {z = x + j y : |y| < x < 1 or |y| < x − 1 when 1