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.