Greenwood press

COMPLEX NUMBERS
17

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Complex numbers in real life

Complex impedance in circuits

Generation of fractals from complex numbers

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CONIC SECTIONS
In the third century 
B
.
C
., the Greek mathematician Appollonius wrote a set of
books dealing with what he called conic sections. He provided a visualization of
ellipses, hyperbolas, and parabolas as intersections of planes with cones. Unlike
ice cream cones, Appollonius’s cone looked like two cones sharing a common
vertex. The picture on the left shows the parabola that is formed by cutting the
cones with a plane parallel to the slant of the cones. It took almost 2,000 years
before applications of conic sections emerged in science and engineering, but
they are now all around us. The middle picture shows a microwave antenna. The
microwaves emerge from the transmitter outside of the reflector at its focus. The
reflector concentrates the wave, as shown in the right-hand picture. Without the
parabolic reflector, the waves would dissipate following the inverse square law.
(See Inverse Square Function for more information.)
The picture on the right shows how the waves from the energy source emerge
from many directions. The energy source is positioned at the focus of the
parabola. Once the rays hit the parabolic reflector, they are transmitted out in par-
allel direction. This concentrates the energy in one direction. For this reason, the