The term ‘nonlinearity’ here means that there is an exponent other than one on the dependant variable. In the above hyperbola-ellipse example, when we take the two square roots, if complex numbers are generated, there are actually four values output for each single input. These are the real and imaginary parts of each root.
So, the hyperbola-ellipse equation:
with a = b = 1 and C = 1, and rearranged, is:
and, in 3Di becomes:
In this form there are two output values –one real and one imaginary – for each root (function) produced for each input value.
Both of these roots are then graphed in accordance with the 3Di axes: The input being on the horizontal axis. (Left middle finger, pointed right.) The real output being on the vertical axis. (Left thumb, pointed up.) And, the imaginary output on the ‘depth’ axis. (Left index finger, pointed forward.)
This then, gives the two separate functions generating two three-dimensional space curves:
As we go on, we shall see that whenever roots are taken, multiple 3Di functions are generated. This provides a tool for the analysis of non-linearity; meaning exponents on the dependant variable of any given function.
The two functions given by:
by adding the notation of ‘de Moivre Numbers’,
where is the root of the roots of unity,these two functions can be written,
in which means both second roots of And individually:
So, with a third degree exponent on y:
And, means all three cube roots of
with: (first root in blue, second in red and third in black),
note that the segments of the three functions that are on the FRP are the upper half hyperbola part of the black graph, and the lower ellipse part of the blue graph.
Animation 4 <’Conic Nonlinearity’
For any degree on the dependant variable:
the individual roots functions are:
in which are all of the roots of for any degree In section 4.56, Lame Curve Bifurcation, we will see that the exponent on may be any degree also.
See also: section 7.1 Helix and Spiral Non-linearity, section 4.11 Quadratic Nonlinearity, and section 4.64 Polynomial Nonlinearity.