Assignment 7: Tangent Circles and the Ellipse
By: David Drew
EMAT 6680, J. Wilson
††††††††† We begin this investigation by saying that many things in this world are guided by conic sectionís principles. And as we remember conic sections are defined to be circles, hyperbolas, ellipses, and parabolas. There are conic sections in virtually all aspects of our daily lives. Parabolas are the main method for making headlights in our cars, and binoculars to watch football games with. And ellipses are the guiding force that makes our world rotate around the sun without shooting off into space, which makes this section the most interesting. Iím basically going to take a tweaked view of problem 6, which investigates the locus of the centers of the constructed tangent circles with two given circles one on the inside of the other.
We will now start by a creation of a circle that is tangent to two other circles one being on the inside of the other.
1. Start by making two distinct circles one inside the other and without them touching.
2. Next we need to mark off a point on the larger circle where we can construct the smaller circle with the radius on the larger circle. And then we make a ray from the center of the larger through the center of our constructed circle, with the intersections of the ray and the constructed circle labeled.
3. From here we take the intersection on the outside of the large circle and we connect it with the radius of the original smaller circle. After weíve made this segment we find the midpoint and drop a perpendicular of the segment through the midpoint. And hopefully this perpendicular will intersect with our ray.
4. If we hide everything and construct another circle at our newest intersection point and we have the radius extend to our point on the larger circle we should get a circle that is tangent to our first two circles.
Our construction turned out to be correct and if youíre not convinced and youíd like to investigate then be my guest and click here.
Warning: Hereís where our investigation takes an elliptical bend (no pun intended).
If we stay with our construction from above then all we need to do is construct a locus by selecting the point on the edge of our larger circle followed by the center of our tangential circle. And the locus should look like this.
Now the trick is to prove that this construction is in fact a real ellipse. We need to do this because although we may trust GSP and our eyes we still may see something that actually isnít there at all. So letís begin with the proof. First we need to label our points to make an elegant and concise proof.
If we say that A and B are the foci of the ellipse then by the definition of an ellipse the distance from A to some point C on the ellipse plus the distance from B to the same point C is a constant. We begin the proof by saying that BD is a constant because it is the sum of BF (our large circleís radius) plus FD (our small circleís radius, which is equal to GA). So the first part of our ellipse, BC, is just BD minus CD, which turns out to be the radius of our smaller circle plus the radius of our tangential circle. This length, CD, is not constant, but fortunately for us it is congruent to some other piece of our picture. It turns out that triangle ACE is congruent to triangle DCE, and Iíll show you why. We constructed segment AD, found the midpoint, and called it E. So E bisects AD and therefore AE is congruent to DE. We also constructed the line through EC by dropping a perpendicular with AD through E, which makes both angles AEC and DEC right angles, and hence these angles are congruent also. Finally both triangles share a side CE. So by the side-angle-side axiom we can say that triangles ACE and DCE are congruent. Therefore AC must be equal to CD. And as we said before BC plus CD is a constant and by our definition of an ellipse, and thanks to the SAS axiom we can say that AC + BC is a constant length.† Through all this we learn that our locus of construction is in fact an ellipse as we had theorized.
The following picture is what happens it our smaller circle moves out of interior of our larger circle. Can you guess that happens to the locus without looking at the picture?
If you guess that it would create a hyperbola then you guessed right. Or it could be the case that you saw the picture before you even read the question. Whatever the case may be, the real question comes as an aside to the above proof. Can you prove that the locus created here is in fact a hyperbola? I wonít give a rigorous proof here, but remember that the constant is not a sum but a difference between two points on the hyperbola, and if you include an absolute value sign then youíll get both branches just like our picture. You can mess around with that construction here.
To conclude our discussion on the Elegant Ellipse I have included a GSP file of the Earth rotating around the Sun. This is really how the orbit works, and how many orbits work through out our solar system. The Sun acts as a foci and the other foci is somewhere out in space. Whatever the case may be itís fun to watch this demonstration so click on Earth and Sun to view it. Also click on Gears to view a GSP demonstration of how elliptical gears work.
Write Up by David Drew. All work is credited to the author who is particularly happy with the Elliptical Gears Construction.