The Transcomplex Trigonometric Sin Function
The transcomplex sin function is:
Function
domain for this example: A complex rectangle of 4 positive radians on the X-axis and 2
positive radians
on the iZ axis with 4 subdivisions per unit, the Y~-axis going
through the center of the rectangle as show in the figure at right. 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 the
rectangular domain 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 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. Note how the surface generated by this strip wraps around the function subdomain.
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 a strip of the rectangular domain defined in
Fig. 1. The strip goes from X = 1 to X = 2 all the way along parallel to
to the iZ axis.
Figure 1-d shows the surface map produced by the positive side
of the complex domain of the plane X-iZ.
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 sin function. Now we see that this figure is just the planar
projection of a more general function 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
View: Top. The observer is looking from the positive Y~-axis
toward the origin of coordinates. This is the same Fig. 1 seen from above.
Picture
View: Frontal. 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.
Picture View: Axonometric. 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.
Figure 6 is the same
Fig. 5 seen from the iZ axis toward the
origin. Now it can clearly be seen that the real function y = sin(x) is a very
special mapping of the transcomplex sin 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.
Below is shown the resulting transformation of the transformation of Da Vinci's Mona Lisa under the transcomplex
trigonometric sin map.
