an Exploration of Tangent Circles

This exploration will investigate the characteristics regarding the locus of points for different cases of the tangent circles disscussed on Dr. James Wilson's Assignment page: "http://jwilson.coe.uga.edu/EMT668/Asmt7/EMT668.Assign7.html"

The different cases for constructing the tangent circles are:

a. the smaller circle is external to the tangent circle.
b. the smaller circle is internal to the tangent circle.

In both cases, the construction of the tangent circle is similar, only differring in the location of the segment j and k as shown in the sketch. The circle on the left is case b. and the circle on the right is case a.

A good question would be, how does this affect the placement of the tangent circle?

After observing both cases, and trying to determine the relationship between the locus of points and the location of the smaller circle with respect to the tangent circle, I had found that the locus of points is primarily determined by the location of the tangent circle (shown in red on the above figures). When the tangent is interior to either circles then the locus of points constructs an ellipse, whereas when the tangent is exterior to either circles then the locus of points constructs an hyperbole.

The light green collection of points represents the locus of points for each case. The left side represents the tangent circle interior to the parent circles, while the right side represents the tangent circle exterior to the parent circles.

This is a very problem rich environment of which I will only attempt to prove why the locus of points behave as either an ellipse or a hyperola.

Before I start the exploration, I will need to define ellipse and hyperbola:

Merrian Webster defines an ellipse as, "a closed plane curve generated by a point moving in such a way that the sums of its distances from two fixed points is a constant"

Merrian Webster defines a hyperbola as, "a plane curve generated by a point so moving that the difference of the distances from two fixed points is a constant."

I have highlighted fixed points, and constant in that they will be crucial to my attmept to prove that the locus of points for each case is indeed either a hyperbola or an ellipse.

Please refer to the figure below for my first attempt:

Since the Tangent circle (shown in red) is exterior to the parent circles, then we can claim that the locus of points (the trace of K as L follows circle D), will be a hyperbola. By the definition of a hyperbola, we need two fixed points and the difference of the points from the fixed distances is constant.

Triangle FDK is highlighted to better illustrate where the focii are and the relationship of the two fixed points (D,F). For this proof we want to show that points F and D are the fixed points are their distances will remain constant when point L is rotated about circle D. The goal is to show that DK- FK = R+r

Segment DK can be defined as the sum of the radius of the large circle (R) , with the radius of the small circle (r), and the segment MK.

DK = R+ r + MK (Equation 7.1)

Since triangle MKF is an isosceles triangle,

MK = FK

Segment FK can de defined as the difference of radius of the tangent circle LK and the radius of the small circle (r).

LK- r = FK (Equation 7.2)

Taking the difference of equations 7.1 and 7.2 we get:

R + r + MK - (LK - r) = DK - FK

R + r + MK - FK

since FK = MK,

R + r = DK - FK

Note: This is true when the small circle (circle F) is external to the tangent circle, but when the small circle is internal to the tangent circle, the constant will be of a different value (R - r) for the fixed distances.

Since the Tangent circle (shown in red) is interior to one of the parent circles, then we can claim that the locus of points (the trace of B as L follows circle D), will be an ellipse. By the definition of a ellipse, we need two fixed points and the sum of the points from the fixed distances is constant.

Triangle ABC is highlighted to better illustrate where the focii are and the relationship of the two fixed points (A,C). For this proof we want to show that points A and C are the fixed points are the sum of their distances will remain constant when point F is rotated about circle A. The goal is to show that AB+BC = R-r

Segment AB can be defined as the difference of the radius of the large circle A (R), small circle F (r), and segment BG

AB = R - (r + BG) (Equation 7.3)

Since triangle CBG is an isosceles triangle,

BC = BG

Segment BC can de defined as the difference of radius of the tangent circle BE and the radius of the small circle (r).

BC = BE - r (Equation 7.4)

The sum of equations 7.3 and 7.4 we get:

R - (r + BG) + BC = AB + BC

R - r - BG + BC

since BC = BG,

R - r = AB + BC

Note: This is true when the small circle (circle C) is internal to the tangent circle, but when the small circle is external to the tangent circle then the value of the constant will be R + r for the fixed distances.