Some functions are more easily described in terms of angle and radius. The angle is the independent variable, and the radius at that angle is the dependent variable. Polar plots are widely used in electrical engineering to describe the radiation pattern of an antenna. Some antennas are designed to transmit (or receive) electromagnetic radiation in a very narrow beam. The beam shape is known as the radiation pattern. One way to achieve a narrow beam is to use an array of simple dipole antennas, and carefully control the phase of the signal fed to each antenna. In the following example, we will consider seven short dipole antennas set in a straight line:

# A linear broadside array of short vertical dipoles 
# located along the z axis with 1/2 wavelength spacing
var('r, theta')
N = 7
normalized_element_pattern = sin(theta)
array_factor = 1 / N * sin(N * pi / 2 * cos(theta)) \
/ sin(pi / 2 * cos(theta))

array_plot = polar_plot(abs(array_factor), (theta,...