Assignment 10

Exploring Parametric Curves

Drew Wilson

In this exploration we will explore the parametric equation shown above for different values of a and b.

First let's notice that the x values are determined by the function cos(t) and the y values are determined by the function sin(t). Based on our knowledge of these two functions we know that when t=0 or 2π cos(t)=1, when t=π cos(t)=-1. We also know that when t= or cos(t)=0. Now, for the sine function we have the opposite. When t=0, π, or 2π, then sin(t)=0, and when t= or , then sin(t)=1 and -1 respectively. The same occurs for both functions for all multiples of each factor, for example for all multiples of t=2π will result in cos(t)=1. This is why the cosine and sine functions are used to represent circles, because multiples of each input have the same output, this is an example of how many revolutions the circle has made.

Now that we know how the sine and cosine functions operate, let's make some inferences about what may happen for different values of a and b. For the x-values, we can make the assumption that when t=0 or 2π the value of cos(t)=a because we know for these values of t, then cos(t)=1 and 1 is the multiplicative identity, so we know that a will be the x-value. Similarly, if t= or then cos(t)=0, so this will have no affect on the y-values because x=0 for these values of t. Now if we look at the y-values and the sine function, we will notice that when t= or then sin(t)=b and -b respectively because we know sin(t) for these values of t is 1. Similarly, when t=0 or 2π then sin(t)=0, so for these values of t the y function isn't affecting the x-function becaus the y-function is zero. Now we can make a conclusion based on these ideas. The range of x-values will be determined based on the value of a, and the range of y-values will be determined based on the value of b. If a=2, then the x-values will range from -2 to 2, if b=2, then the y-values will range from -2 to 2.

There are three different cases to consider when dealing with the parametric equation. The first case would be when a=b, and the second case would be when a ≠ b. The third case is when a=b=0.

In case one, we will notice that when a=b, the parametric equation is a circle. This is true because our x-values and y-values have the same range (x-values -a ≤ x ≤ a, y-values -b ≤ y ≤ b). The points in between will follow for a circle because we know the x values of the unit circle is determined by the cosine function and the y values of the unit circle are determined by the sine function. Similarly, this parametric equation has the x values determined by the cosine function and the y values determined by the sine function, so we know that when a=b, we will have a circle.

In case two, we will notice that when a ≠ b, the parametric equation is an ellipse. If we consider the case when a=2 and b=1 and notice the range of x-values and y values: -2 ≤ x ≤ 2, -1 ≤ y ≤ 1. Since the difference in the range of x is a total of 4 units and the difference in the range for y is a total of 2 units, we know that this equation would not represent a circle because the radius isn't the same. The center would be at the origin and we would have a radius of 2 on the x-axis and a radius of 1 on the y-axis. This would give us an ellipse with the major axis of the ellipse represented by the x range and the minor axis of the ellipse represented by the y range. Similarly if we consider the values a=1 and b=2, then we would have an ellipse with the major axis represented by the y-axis with range -2 ≤ y ≤ 2 and the minor axis represented by the x-axis with range -1 ≤ x ≤ 1. Based on this information, we can tell which axis is the major axis by the values of a and b, whichever value is greater will represent the major axis of the ellipse because the range of values will be greater. Below are a couple of examples of possibilites. The picture to the left is when a=b=2, the picture in the middle is when a≤b (a=1, b=2), and the picture on the right is when a ≥ b (a=2 b=1).

In case three, when a=b=0, this will represent the origin because the x value is zero and the y value is zero, therefore the origin is represented by the polar equation with these parameters.

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