Two Circles Hoola-Hooping
An Exploration of Tangent Circles
This exploration is based off of the prompt given in Exploration 7 which says:
Given two circles and a point on one of the circles. Construct a circle tangent to the two circles with one point of tangency being the designated point.
The prompt goes on to explain the different situations involving finding these tangent circles. There are actually two types of tangent circles for each of the two given circles (if one of the circles is inside of the other): the tangent circle lying inbetween the two circles and the tangent circle enclosing the smaller circle:
There are also three different scenarios to consider: when one circle is located inside of the other, when the circles intersect, and when the circles are disjoint or completely separate from each other
What are some initial properties we see when we construct these types of circles? First is the number of possible tangent circles that can be created as the point of tangency (blue point) is moved along one of the circles. In the first scenario, where one circle is located inside the other, there is an infinite number of circles.
This is because there is always a diameter created between the two circles, whether this tangent circle’s diameter contains the diameter of the smaller circle or not.
The only situation where this would not occur is when the smaller circle inside is tangent to the other circle. Therefore, there would be no way for there to be a tangent circle at the point that the circles share.
The next situation is when the circles intersect. We have already seen the scenario where the two original circles are tangent to each other. This same scenario can happen with the second circle on the outside of the other circle.
But what if this is not the case? What if the circles simply intersect and are not tangent to each other? This would mean that they circles have two intersections. We would still encounter an issue at the intersection points. Since, once again, the circles share these points, no tangent circle can appear.
The third scenario is when the two circles are completely disjoint. Therefore, they share no points and will not encounter the scenario shared above.
So, if this limitation does not exist, should there be an infinite number of tangent circles? This is what one would expect. However, there is exactly one situation for this scenario in which a tangent circle cannot be formed, and that is when a blue tangent point is in a position where only a tangent line can be formed.
No circle can be formed that is tangent to both circles. To be able to create a circle, it would have to intersect instead of simply touch at a point. In fact, this is the point where the tangent circles go from enclosing one of the circles to enclosing the other, as the images above display.
What would happen if we animated the blue tangent point while tracing the midpoint of the tangent circle? We will have to examine this scenario by scenario again. Let’s begin with when one circle is inside another. We would obtain a shape similar to the ones below, depending on the size and location of the circles:
So, we know that this will always produced a closed curved figured that is elliptical in shape. Will this always be the case?
What if the two circles shared the same center?
We would then have a circle instead of an ellipse. This makes sense because the tangent circle would never change size. Thus, it just follows a circular path around the two circles.
Let’s now analyze the situation where the two circles intersect at two points. What kind of shape would be made if we did the same animation?
It looks like we have the same situation as before, except there are no trace points at the intersection, which we have discussed previously. Is there a situation where we can form a circle? No, because there is no scenario in which we could have one of the circles fully inside of the other if they are to intersect like this. Even when we give the two circles the same diameter with the edge of each intersecting the center of the other, we still form an ellipse with two missing points.
Why do these particular animations form ellipses? It is because as the intersection (blue) point travels around one of the circles, the tangent circle begins to vary in size. When the two circles do not share a center, the circle inside is closer to one side of the circle than the other. At the close side, a very small tangent circle is formed. On the farther side, the tangent circle is much bigger. When the blue intersection point is on either of these sides, the size of the tangent circle does not change as drastically. This causes the trace to look very rounded, almost like a circle (the edges of our ellipse). When the blue point transitions from one of these sides to the other, it must change size dramatically to go from being very large to very small, or from being very small to very large. This drastic scaling either pulls the center of the tangent circle in (large to small) or pushes the center out (small to large). This is why the traces here are longer.
Let’s take the situation where the circles are disjoint. We saw before that there was one point for the blue intersection where a circle could not be formed, only a tangent line. Let’s do the same animation we did for the previous scenario:
It looks like that in this case, the traces form a hyperbola. But at what point do the traces switch from one side of the hyperbola to the other? Well, as we discussed earlier, there is a point where only a line can be formed. This is where the circles switch from enclosing one circle to enclosing the other. This is the point where the sides of the hyperbola switch.
Why do these form a hyperbola instead of an ellipse? As previously stated, we have two situations for these types of circles: one of the circles is enclosed by the tangent circle or the other circle is enclosed. These two types are separated at the point at which the tangent line appears. In essence, we are taking the tangent circle and circling it around one of the circles, and at the tangent line point, we switch to circling it around the other circle. This is why there are two separate traces going in two separate directions.
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