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
Read more about this topic: Motor Variable
Famous quotes containing the words elementary, functions, motor and/or variable:
“When the Devil quotes Scriptures, its not, really, to deceive, but simply that the masses are so ignorant of theology that somebody has to teach them the elementary texts before he can seduce them.”
—Paul Goodman (19111972)
“The mind is a finer body, and resumes its functions of feeding, digesting, absorbing, excluding, and generating, in a new and ethereal element. Here, in the brain, is all the process of alimentation repeated, in the acquiring, comparing, digesting, and assimilating of experience. Here again is the mystery of generation repeated.”
—Ralph Waldo Emerson (18031882)
“The motor idles.
Over the immense upland
the pulse of their blossoming
thunders through us.”
—Denise Levertov (b. 1923)
“Walked forth to ease my pain
Along the shore of silver streaming Thames,
Whose rutty bank, the which his river hems,
Was painted all with variable flowers,”
—Edmund Spenser (1552?1599)