Department of Mathematics Education
J. Wilson, EMAT 6680
Matthew Tanner
EMAT 6680
Write-up #11
July 16, 2004
Like the Cartesian
coordinate system, polar coordinates offer another unique description of the
location of a point in space. Whereas
points referred to using a Cartesian coordinate system refer to an original
point and n vectors meted
parallel to axes called bases, points referred to using a polar coordinate
system refer to a distance from an original point according to a deflection
from n-1 reference
vectors. Sets of points described
simply using a Cartesian system are frequently cumbersome and protracted
using a polar system and vice versa. In this study, simple
polar equations of the form r = a + b cos(kӨ) are investigated.
Figure 1 Referring
to a Cartesian analogy of the graph of a polar equation, we see that within
the range of 2p, the graph completes 5 cycles because of the scalar multiple of q. The
difference between the maxima and the mean of the graph (1 for the cosine function without a scalar)
is 3. The mean amplitude of the graph (0
for the cosine function) is 7.
Figure 2 Polar graphs of equations of
the form r = b cos(kq) do not have a shift in the mean amplitude of the cosine
function. We observe that when k is an even integer, the number of
“petals” is double – half correlating with the positive semi-cycles of the
Cartesian graph of the cosine function and half correlating with the negative
semi-cycles.
Figure 3 When
k is an odd integer, we note
that the number of petals is not doubled.
This is due to the graph of the function between 0 and p coinciding with the graph of the function between p and 2p. |