Parametric Equations Exploration
Suppose we have the following equations:
For simplicity, let’s investigate the graph for values of a = b and t-values between zero and 2p.
For values of a = b greater than the absolute value of 1, we will get a circle of radius 1 (not =0). For instance, the following graph represents a = b = 3. Because we are dealing with a circle, it will not matter if we use t-values larger than 2p since the graph will continue along the circle.
Now we can consider rational values of a and b.
For a = b =.5 and 0<t<2p, we only see the top half of circle located in quadrants I and II.
However, if we consider larger values of t, we will get slightly different results:
For a = b =.5 and 0<t<3p
For a = b =.5 and 0<t<4p
As you can see, as we increase the range of t-values to include larger multiples of Pi, we will again arrive at a circle. If we try larger values of Pi, we will continue to get circles because the graph will repeatedly create circles.
For a = b =-.5 and 0<t<2p, we see only the bottom half of the circle in quadrants III and IV.
Again, if we increase the t-values to larger intervals of Pi, we will again arrive at a circle.
a = b=-.5 and 0<t<4p
For a = b =.25 and 0<t<2p, we only see ¼ of the circle in quadrant 1
Likewise, for a = b = -.25, we see ¼ of the circle in quadrant IV.
This time, it will require us to use 0<t<8p to get a full circle:
Quite predictably, for a = b =.75 we see ¾ of circle in quadrants I, II, and III.
Finally, if a = b = -.75, our resulting circle resides in quadrants II, III, and IV.
Lastly, we can use 0<t<3p to get a full circle for the above values of a = b = .75 or -.75:
For a ¹ b, the graphs become quite interesting. However, the t-values seemingly do not affect how many “loops” each of the following graphs. For simplicity’s sake, we will maintain the values of t to remain between 0 and 2pi.
For a = 1 and b any natural number, the number of “loops” corresponds to the value of b. For instance, in the following graph, for a = 1 and b = 4, there are 4 “loops”.
As b increases, so do the number of “loops”.
Suppose we consider large values of b. For instance, let a= 1 and b = 15. According to the pattern seen thus far, we should see 15 “loops”.
As predicted, we see 15 “loops”. What about a = 1 and b = 50?
Please note that for all of these values, the graph oscillates between -1<x<1 and -1<y<1.
However, if we explore negative values of b, we see the same picture.
What about rational values of b? For values between 0 and 1, the graph appears to behave much like a slinky. Again, we have oscillatory motion for -1<y<1 and -1<x<1. As b approaches 1, the graph appears more like a circle and will eventually be a circle for b = 1.
Suppose we change the values of a and keep b = 1. For a values between 0 and 5, the graph follows a similar oscillatory pattern except in a vertical fashion. These graphs also have a domain of -1<x<1 and a range of -1<y<1. Obviously, the values of a and b control which direction the graph oscillates. Unlike the values of b, the number of waves also corresponds to the value of a for odd values. For even values of a, we see the period corresponds to half the value of a.
For instance, suppose a = 5 and b = 1. We can predict a graph with vertical oscillations and 5 “loops”.
Now suppose a = 10 and b =1. We can predict another graph with vertical oscillations and 5 periods.
To see other values of a, Click here for animation. This animation has 0<a<6, b=1, and 0<t<2Pi.
Perhaps some of the more interesting observations are when a is a whole number, the graph is continuous. When a=1, we again see a circle; if a =2, we see a sideways parabola; if a=3, we see a vertical “spring” with 3 “loops”; if a=4, it resembles a=2 with an extra curve; and a=5 has 5 “loops”. It appears that for odd-values of a, we will see a figure that is “closed” meaning that it will continue along that path without having to move backwards. For instance, an object can continue to travel along a circular path forever.
However, for even values of a, we will result in a graph that is not “closed”:
When we use negative values of a, we get similar results as when b was negative. In fact, the graph is identical to the resultant graph from positive values of a. Essentially, we are only interested in the absolute value of a since the graph is the same for negative values of a.
For rational values of a, we see similar oscillatory motions except that the movements are horizontal. Furthermore, all non-whole values of a will result in a graph that is not “closed”: