A parametric curve in a plane is defined as continuous functions in the form:
These two equations are called the parametric equations of that curve, and define the ordered pairs (x,y). The extent of the parametric curve will depend on which range of t we choose. By using Graphing Calculator 3.2 we can explore various types of parametric equations. Let's begin by look at the following equations:
Since we know both continuous functions, sin(x) and cos(x), have a period from 0 to 2pi. So, let's choose our parameter t similarly:
These parametric equations define a set of ordered pairs (x,y) over an interval t. The graph of these parametric equations over the interval looks like the unit circle, with a domain of [-1,1] and a range of [-1,1]. To see the importance of an appropriate interval for t, let's look at the same equations over different intervals of t. What would happen if t did not go all the way to 2pi?
We can see from the above images that if we define t over a shorter interval, our circle will not be complete. The graph of our parametric equations starts at point (1,0) and traces a circle in a counter-clockwise direction, but the circle is not closed. Here, our continuous functions will only be graphed with respect to the given interval t, and will only be closed if we choose and appropriate interval.
Now let's investigate various graphs for the parametric equations:
We have already seen this graph above when a=1 and b=1. So, let's look at different values for a and b. There are immediately three different types of values for a and b we can investigate: when a < b, when a = b, and when a > b.
There seems to be a very specific relationship between the values of a and b. Each graph above is symmetric about the x-axis and the y-axis. When a = 1, it seems that b is equal to the number of "loops" shown in the graph. View the animation when a=1 and our value for b goes from 2 to 20. Notice what happens to the number of loops graphed.
It seems that when a is an even value, and b is odd as shown above, our graph does not appear closed. Also, each graph shown above is only symmetric about the x-axis. View the animation when a goes from 0 to 9 and the value of b is fixed at 10. Is seems as though when the value of a is less than the value of b, the graphs appear to be expanding and contracting in the horizontal direction, or along the x-axis. In what ways does the graphs generated here look different from the animation above?
What can we conclude for a < b?
By varying our constants a and b, our graphs will still have a domain of [-1,1] and a range of [-1,1]. Each graph generated will be symmetrical about the x-axis. Only certain graphs will also be symmetrical about the y-axis.
View the animation when a=b going from 0 to 200. Notice the appearance of our circle as a and b change together. Why does the graph appear to be getting thicker?
What can we conclude for a = b?
Again, our domain and range have not changed by varying the constants a and b. As long as a = b, the graph generated will result in a circle of radius 1. When the values of a and b are increased, our circle seems to be getting thicker. This is because our interval t has not changed. So, as the value for a and b increase, our circle is being traced over more and more times. If a and b are negative values, their graph appears the same as if they were positive values. This is because our parametric equations are still being graphed over the same interval, and so our circle will be always be closed. The negative value simply changes the direction of the 'trace' of our parametric equations. In other words, as shown above in "The Importance of t," our a and b values both equal 1. The images of varying the interval of t show that our graphs begin at point (1,0) and the circle is traced in a counter-clockwise direction. A negative value for a and b simply mean the graph of our parametric equations was again started at point (1.0), but was then traced in a clockwise direction. Once the interval is such that our circle is closed, we are no longer able to see the difference between the directions in which our circle was traced.
There seems to be a very specific relationship between the values of a and b here as well. View the animation when b=1 and our value for a goes from 2 to 20. Notice what happens to the number of loops graphed.
When a is larger than b, it seems as though the graphs are expanding and contracting. View the animation when b goes from 0 to 9 and a is fixed at 10. In which direction do these graphs appear to be moving?
What can we conclude for a > b?
Again, our domain and range have not changed by varying the constants a and b. Each graph generated will be symmetrical about the x-axis. Only certain graphs will also be symmetrical about the y-axis.
The types of curves that have been generated by our original parametric equations compare with what are called Lissajous Curves. These types of curves are generated by a specific formulas, similar to the ones we used above. For more information and to experiment with Lissajous Curves, visit the following websites:
Key Curriculum Press, Lissajous Curves Math World, Lissajous Curves
Check out more interesting graphs for various values of a and b!