Polar Equations

The following equation will be investigated:

r = a + b cos (kJ)

The first several graphs have a fixed at 0, b = 1 and variations with n: 0+ 1*cos(nq).

n = .01

n varies from > .01 to < 1.0 in the next 4 pictures. It shows how the circle at n = .01 'breaks' apart and begins to form 'loops', then recombines to form more than one circle at n = 1.

n = 1

As n increases, the circles at n = 1 split apart and form lobes.

n = 2 here, notice that with n = 2 , you get 4 lobes; what is interesting is...

...at n = 3, there are 3 sets of 2 lobes superimposed, not 3 lobes. Because...

as we will see, with an even number the pattern is 2n. Now, at n = 4 the result is 8 lobes or 2n lobes.

Here n = 9 (double lobes).

Finally, n = 10 with the result of 20 lobes (2n lobes).

It is interesting that when n varies from -10 to -.01 that one obtains the identical results in each case as those above.

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This equation is: 1 + 1*cos(n*q), with n beginning with 1. The following graphs occur when n moves from 1 to 10:

n = 2:

n = 7:

n = 8; There is 1 lobe for every increment of n (1n = # lobes). Again, this was for 1 + 1*cos(nq).

When the values for n vary from -10 to 0, the identical results occur as above.

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The next series begins (and ends quickly) with a + 1*cos(0*q):

As a increases from 0 to 100 with 1*cos(0), [of course, cos(0) = 1],the figure remains a circle with a diameter that equals 2a + 2 , therefore, a + 1 = the radius. Here, a = 10:

a = 100:

When n varies from -10 to 0, the result are circles; when n = -1 the circle disappears as expected; when n < 0, r = a + 1 but with a diameter of 2 less than circles with n > 0. Here n = -10:

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This series is for r = n + 1 cos(1q)

When n = 0, the figure is a circle; a cardioid when n = +/-1; For -1 < n < 0 and 0 < n < 1 the type of figure is :

For the above graph, n = +/- .2; below, n = +/- .6:

The figure inside the larger circle is 'moving out' from the inside, making a cardioid at n = +/- 1 and then forming a circle (proper) at n = 3. Up to n = 3 it is still an imperfect 'looking' circle.

For the series of r = 0 + n cos (1q), when n = 0 there is no circle of course, but for n > 0, a circle forms such that its left side stays fixed at (0,0). For n < 0 the figure is a circle but it is fixed on its right side at (0,0):

Finally, with the series of r = 1 + n cos (1q), a circle is formed at n = 0, a cardioid at n = 1, for n > +1 or n < -1, a circle forms from the cardioid and gets larger inside a larger circle:

It is interesting that both circles are fixed at (0,0) and the 'outside' edge of the inner circle versus the outer circle are 2 units apart consistently (even when n = 100). Here, the inner circle reaches -9 and the larger circle reaches -11 (n-1 and n+1 for values of n).

When cos is replaced by sin in the preceding graphs the circles are oriented 90 degrees versus the previous graphs. With n > 0, the double circles are on top of the x axis and under the x axis with both circles fixed at (0,0).

Further investigations are left up to the reader.