CIRCLE AND ELLIPSE

By Dario Gonzalez Martinez

We will start considering the following parametric equation:

Our objective is to explore what curve (or curves) is associated to this parametric expression. We will explore how the variation of parameters a and b affects the shape of the curve related with the parametric expression. To this end, we will observe that the analysis is reduced to two board cases:

1) What is the shape of the curve when ?

2) What is the shape of the curve when ?

CASE 1:

For this case our initial expression is reduced to

We will attempt an initial graph for that expression when a = 1. This case is shows in figure 1 below:

Figure 1

Figure 1 supports our perception that the curve should be a circle with center in the origin of the system and radius 1. This conjecture will become stronger if we observe the animation 1 below which shows how the shape of the curve change according to the variation of parameter a.

Animation 1

The above animation suggests that when we vary parameter a, the shape of the curve is the same, a circle, the only difference is this circle has different radius, which seems to be equal to the value of a. Although we just made a conjecture, it is possible to prove that our conjecture is true. Let’s take a look at the figure 2 below:

Figure 2

The circle with center O is a circle with radius a. We will denote with t the angle formed by segments a and x, and segments x and y are perpendicular to each other. We can use trigonometric relations to obtain

We also consider the trigonometric relation

By replacing the above relation in the latter one, we have

This expression represents the equation of a circle with radius a and centered at the origin of the system. So, our conjecture is actually true.

CASE 2:

First, we will assume that a > b, so we start trying a = 2 and b =1 to graph an example that guide our analysis. In other words, we will consider the expression

In this manner, the curve will look like that in figure 3 below:

Figure 3

Figure 3 above suggests several interesting facts. The curve associated to the expression seems to be an ellipse which has major axe equal to 2a = 2(2) = 4 and minor axes equal to 2b = 2(1) = 2. To make us sure about it, we should observe the following animations:

Animation 2(a)

Animation 2(b)

The above animations show how the curve behaves when parameters a and b vary. Animation 2(a) shows the variation of parameter a and b fixed, whereas animation 2(b) shows the variation of parameter b and a fixed.

Once again we can show that our conjecture is actually true. Consider the figure 4 below:

Figure 1

The circles blue and green are concentric. A and B represent the intersections points of line OA and the blue and green circles, respectively. Lines AC and BD are perpendicular to x-axis, and the line defined by B and point (x , y) is parallel to x-axis. If we rotate line OA around center O, point (x , y) draws an ellipse with major axis the radius of the blue circle and minor axis the radius of green circle. We will consider as parameter t the angle COA.

From right triangles OAC and OBD we obtain

Therefore, the parametric equation for the locus of point (x , y) are

On the other hand, we know that

Which can be written as follow

From which in turn we can obtain the canonic equation of an ellipse

Thus, our conjecture is true, and the curve for the expression

Indeed represents an ellipse centered at the origin system with major axe equal to 2a and minor axe equal to 2b.

We could elaborate similar arguments to show that the expression

Is also an ellipse centered at the origin of the system when a < b. The only difference is that this ellipse will be “taller” than “longer”. Figure 5 below shows an example for a = 1 and b = 2:

Figure 5

FURTHER ON THE ANALYSIS

Now we will consider the following parametric equation

We will start considering a = b =3, and we make h varies. We can observe the behavior of the curve in the animation below:

Animation 3

We can see that there is an instant, that is, a value of h when the curve becomes into a line segment which passes through the coordinate system origin, but the question is when this occurs. If we think about it we can conclude that a line segment that passes through the coordinate system origin represents a proportional relation between x and y. In other words, we have to observe in which moment the system

becomes linearly dependent. If we reorder the system above, we will have

By using the determinant property of a equation system

We can actually generalize this case for a = b = k, and we will obtain that

Thus, when |h| < |k| the curve becomes more and more similar to the circle

The curve will be equal to the curve above when h = 0. On the other hand, if |h| > |k|, then the curve becomes more and more similar to the circle

The curve will be equal to the curve above when h tends to infinite. This analysis makes sense since we could consider the following:

On the other hand, we should also consider the case when a and b are different. The animation 4 below shows the case when a = 3 and b = 2, and we make h varies:

Animation 4

The previous animation shows that the curve is an ellipse centered at the coordinate system origin. Similarly to our previous analysis when a = b, the curve becomes into a linear segment for some h, which divides our analysis “before” and “after” this specific value of h. Likewise we did before, we have to consider that a linear segment represents a proportional relation between x and y, so the following equation system should be linearly dependent

So, we should have that

or, if we want to generalize this case

which means that h has to be equal to the geometric average between a and b. Therefore, we will have two cases “before” and “after” this specific value for h:

	The curve becomes more and more similar to the ellipse: The curve is equal to the ellipse above when h = 0.
	The curve becomes more and more similar to the circle: The curve is equal to the circle above when h tends to the infinite
Table 1