Brenda King
Polar Equations
Introduction:
Polar equations are popular
with students because of the beautiful pictures that can be made. Polar equations consist of coordinates (r, q ). The directed distance, r, is measured from
the origin (O or pole) and a point P on the graph. The directed angle, q, is measured counterclockwise from
the polar axis (a ray in the positive direction from the origin assigned as 0
degrees) to segment OP.
This investigation will vary
the parameters a , b and k
in the polar equations
r=a + bcos
(kq) and r = a+bsin (kq) for 0 < q < 2p and consider graphs without
parameter a.
Changes in parameter a
The following graphs provide
a quick look at what happens when parameter a is
changed while holding the other parameters at a constant 1.
Changing the curves from
cosine to sine rotates the graph by 90 degrees counterclockwise.
The effect of changing
parameter a is to expand or stretches the graph
border outward (larger values) or inward (smaller values). In most cases the
graph is distorted from it’s original shape.
Changes in parameter b
Since some polar graphs have
distinct shapes names are often assigned to identify or classify the shape
itself. For example, the cardioid has a “heart” shape
and occurs when a/b = 1. When the ratio a/b is between 1 and 2, the cardioid shape is turned into a dimpled limacon. In addition, inner loops are created when the
ratio a/b is 2 or larger.
When the value of b is
negative, a reflection across the y-axis takes place.
Change parameter K for r = a
+ bcos (kq)
When
parameter a =any value (a=1 in this
example), there are k-pedals or loops regardless. The symmetry for an odd k is with respect to
the line y = x. The symmetry for an even k is with respect to both the x and y
axis.
Change parameter K for r = bcos (kq)
In the case when parameter a=0,
the number of pedals and symmetry changes.
For ODD values of k, there will be k-pedals or loops and symmetry is with
respect to the x-axis. When K is EVEN, there are 2k pedals or loops and symmetry
is with respect to both the x and y axis.
Graphs of Sine curves
When cosine is replaced with
sine, the curves are rotated 90 degrees.
Here are some last examples
of the effect of parameter a.
