The Circle of Apollonius and Its Applications in

The Circle of Apollonius and Its Applications in Introductory Physics Michael B. Partensky, Brandeis University, Waltham, MA

T

he circle of Apollonius is named after the ancient geometrician Apollonius of Perga. This beautiful geometric construct can be helpful when solving some general problems of geometry and mathematical physics, optics, and electricity. Here we discuss two of its applications: localizing an object in space and calculating electric fields. First, we pose an entertaining localization problem to trigger students’ interest in the subject. Analyzing this problem, we introduce the circle of Apollonius and show that this geometric technique helps solve the problem in an elegant and intuitive manner. Then we switch to seemingly unrelated problems of calculating the electric fields. We show that the zero equipotential line for two unlike charges is the Apollonius circle for these two charges and use this discovery to find the electric field of a charge positioned near a grounded conductive sphere. Finally, we pose some questions for further examination.

Localizing an Object in Space Description of the problem

Once Fred and Jeff1 were vacationing at a summer camp near Landport, ME. In a Scout game their friend Sam, a member of the Green team, was kidnapped by the team of Reds. After his blindfold was removed, he found himself on board of a yacht anchored in the ocean. The night was dark and Sam could hardly see anything but the blurred outline of the city on his left and the lights from three lighthouses. Two of them, say A and B, were on the land, while the third one, C, was positioned on the large island. 104

Fig. 1. The map of the Landport area showing three lighthouses marked A, B, and C. Other notations are explained in the text.

Using the portable photometer from his “Emergency Physics Tool Kit,” Sam found that the corresponding brightnesses were in proportion 36:9:4. Secretly, he used his cell phone to call Fred and tell him what was up. Fred and Jeff immediately Googled the map of the area surrounding Landport and found that the three lighthouses, A, B, and C (see Fig. 1), form a right triangle with its legs |AB| = 1.5 miles and |AC |= 2 miles. The description of the mapped region asserted that the lanterns on all the lighthouses were exactly the same. In a short time the friends knew the location of the boat, and a group of Greens were able to release

DOI: 10.1119/1.2834533

The Physics Teacher ◆ Vol. 46, February 2008

Sam.

1.5

The question: How did the friends manage to find

the position of the boat?

1

Discussion and solution

Being the best math and science students in their class (see Ref. 1), Fred and Jeff immediately figured out that the ratio of the apparent brightness can be transformed to the ratio of the distances. According to the inverse square law, the apparent brightness of a light source (which can be considered point-sized if viewed from the distance r, much larger than its dimensions) is proportional to G/r 2, where G is the intensity of the source. Given that the value of G is the same for all the lanterns and the ratio of the brightnesses is 36:9:4, the ratio of the distances between the boat S (“S” stands for Sam) and the lighthouses is |SA|:|SB|:|SC| = 1:2:3 (as the inverse square roots of the brightnesses). Focusing on the lighthouses, A and B, they realized that point S is one of all possible points P, 2 times more distant from B than from A: |PA|/|PB| = 1/2. This observation immediately reminded them of something that had been discussed in their AP geometry class. At that time, the boys were very surprised to learn that in addition to being the locus of points equally distant from a center, a circle can also be defined through the distances to two fixed points. They rushed to open the file containing their lecture notes and found: “Circle of Apollonius ... is the locus of points P whose distances to two fixed points A and B are in a constant ratio k:

| PA | | PB |

= k.

(1)

For convenience, draw the x-axis through the points A and B. It is a good exercise in algebra and geometry (see the appendix) to prove that the radius of this circle is | xB − x A | R O =k | k 2 −1 |

(2)

and its center is at xO =

k 2 xB − x A k 2 −1

.

The Physics Teacher ◆ Vol. 46, February 2008

(3)

0.5 0

A

B

-0.5 -1 -1.5

-1

-0.5

0

0.5

1

Fig. 2. The Circles of Apollonius for the points A(-1,0) and B(1,0) corresponding to the ratio g = k (right) and

g =1/k (left), with k taking integer values from 1 (red straight line) through 6 (bright green).

Examples of Apollonius circles with the fixed points A and B corresponding to different values of k are shown in Fig. 2. Interestingly, the Apollonius circle defined by Eq. (1) is the inversion circle2 for the points A and B: (xA – xO)(xB – xO) = RO2 .



(4)

This result immediately follows from Eqs. (2) and (3). (Apollonius of Perga [261-190 B.C.E.] was known to contemporaries as “The Great Geometer.” Among his achievements is the famous book Conics, where he introduced such commonly used terms as parabola, ellipse, and hyperbola.)” 3 Equipped with this information, the friends easily solved the problem. First, they found from Eqs. (2) and (3) the radius and the center of the Apollonius circle L1 with the fixed points A and B and k = 1/2: R1 = 1 mile and xO – xA = –0.5 miles. The latter implies that the center O of the circle L1 is half a mile to the south from A. In the same manner, they characterized the Apollonius circle L2 for the points A and C with k = |PA|/|PC|=1/3.. Its radius is R2 = 0.75 miles and the center Q is 0.25 miles to the west from A. After this 105

was done, Fred opened the drawing tool on his laptop and drew L1 and L2 right over the map of the Landport area (see Fig. 1). Now it became clear that Sam must be located at the intersection of L1 and L2 . Only at those two points is his distance to A two times smaller than to B and at the same time three times smaller than the distance to C. They could rule out one of the points, because it happened to be far inland (see the gray circle). They labeled the other point with a large bold “S ” and easily discovered that the boat was anchored approximately 0.35 miles east and 0.45 miles north from A. The team of Greens jumped in their scooters, and soon Sam was freed.

Two Electrostatic Problems Now students can be introduced to another application of the Apollonius circles that at first sight seems absolutely unrelated to the previous one. We note that both problems considered in this section can be solved without even mentioning the circles of Apollonius and, to the best of our knowledge, they have never been discussed from this point of view. However, using the Apollonius approach allows for a very transparent and elegant solution. Equipotentials for the system of two unlike charges

Consider this problem:4 Two unlike point charges q1 and q2 are located a distance 2a from each other. Prove that among the surfaces of equal potential for this system there is a sphere of a finite radius. Find its radius and the coordinates of its center. Find the potential j on the surface of this sphere, assuming that j() = 0. According to Coulomb’s law, the potential of the point charge is

q j(r ) = k , r where r is the distance to the charge and k is the numeric coefficient—the electrostatic constant. We place the charges q1(>0) and q2(