The Transcomplex Quadratic Function
The transcomplex quadratic function is:
The parametric equations are:
Function
domain for this example: A complex square centered at the Y~-axis.
The square is shown 2 positive radians on the X-axis and 2 positive
radians on the iZ axis with 4 subdivisions per unit. The observer is
looking from above the square domain.
Fig.
1 shows the surface generated by the the square domain above. The picture view
is axonometric. The observer is looking from the positive X,
iZ, and Y~ axes toward the origin of coordinates. Note that in this
picture the square domain is dwarfed in comparison the the surface generated.
This is because each subdivision of the domain is squared, and as we move along
the X-axis.
In Fig. 1-a
is showing the map of a unit strip of the domain parallel to the
X-axis. The strip also starts at x = -2 on the X-axis and ends at x
= 2 on the same
axis. The width is one radian along the positive side of iZ axis, that is
0 <= z <= 1. We can clearly see that the negative side of the strip maps
into the negative side of the iZ-axis, and the positive side of the strip maps
into the positive side of the iZ-axis. So, the strip 'moves' smoothly from the
back to the front while its mapping also does the same, but also curving from
right to left.
Fig.
1-b shows the same Fig. 1-a, but now, with the aid of a grid or a
wireframe, we can see the other part of the domain that was not plotted in
Fig. 1-a.
Figure 1-c shows
the mapping of another strip of the rectangular domain defined in Fig.
1-a. This rectangular subdomain goes from x = 1 to x = 2 all the way along parallel to
to the iZ axis. The presence of the wireframe illustrates that the
behavior of this rectangular region is very different from the behavior of the
previous strip. Now everything occurs within their respective quadrants of the
complex plane.
Figure 1-d shows the surface map produced by the positive side
of the complex plane X-iZ. Note that is subdomain is just the half
'positive half' of the original domain. This surface is at the same time 'half'
of the surface depicted in Fig.-1.
Picture
view: lateral. The observer is looking from the positive X-axis toward
the origin of coordinates. This figure is important because it is one of
the classic figures used in textbooks about complex variables when discussing
the complex quadratic function. Now we see that this figure is just the planar
projection of the more general graph depicted in Fig. 1
The same Fig. 2 but now showing in color only the map produced by the unit-width
strip. The other picture wire frame is the remaining on the original
complex domain. This figure is the same Fig. 1-b but looking at it
from the positive X-axis toward the origin.
Picture
3 is a view from the top, that, is, the observer is looking from the positive Y~-axis
toward the origin of coordinates. This is the same Fig. 1 seen from above.
At
right is a frontal view of the quadratic surface. The observer is looking the Fig. 1
from the
positive iZ-axis toward the origin of coordinates. From Fig. 1-d
can be seen that the higher and lower crests are the plots of the domain
extremes.
At
right, an axonometric view of a thin strip this time centered along the X-axis. This is the mapping of a thin strip 0.25
units to the left and to the right of the X-axis. In this case the function
domain is approaching the real X-axis.
Al
right, Figure 6. This figure is the same Fig. 5 seen from the iZ axis toward the
origin. Now it can clearly be seen that the real function y = x^2 is a very
special mapping of the transcomplex quadratic function. When the thin strip
approaches the X-axis we will have the classical real-valued sin function.
At
right is shown the Da Vinci's Mona Lisa which we can taken as the planar domain
for our function under study to see how it is transformed in a point by point
basis. To approximate the painting's resolution, the subdivisions of
the complex domain was incremented accordingly. The domain is positioned
at the upper left corner of the painting; for this reason, some lower border or
right border cropping of the original may occur.
To the right is shown the resulting transformation of the transformation of Da Vinci's Mona Lisa under the transcomplex quadratic map. Note the small outer side of the rectangular domain which is the original Da Vinci's painting.
