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Spatial Localization Problem and the Circle of Apollonius. Joseph Cox1, Michael B

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Spatial Localization Problem and the Circle of Apollonius. Joseph Cox1, Michael B. Partensky2 1 Stream Consulting, Rialto Tower, Melbourne, Victoria 3000 Australia email: [email protected] 2Brandeis University, Rabb School of Continuing Studies and Dept. of Chemistry, Waltham, MA, USA email:[email protected] The 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 mathematical physics, optics and electricity. Here we discuss its applications to “source localization” problems, e.g., pinpointing a radioactive source using a set of Geiger counters. The Circles of Apollonius help one to analyze these problems in a transparent and intuitive manner. This discussion can be useful for high school physics and math curriculums. 1. Introduction 2. Apollonius of Perga helps to save We will discuss a class of problems where the Sam position of an object is determined based on the analysis of some “physical signals” related to its Description of the problem location. First, we pose an entertaining problem Bartholomew the Frog with Precision Hopping that helps trigger the students’ interest in the Ability could hop anywhere in the world with a subject. Analyzing this problem. we introduce the thought and a leap [1]. Publicly, he was a retired Circles of Apollonius, and show that this track and field star. Privately, he used his talent to geomteric insight allows solving the problem in help save the world. You see, Bartholomew had an elegant and transparent way. At the same become a secret agent, a spy - a spook. In fact, time, we demonstrate that the solution of the only two people in the whole world knew who inverse problem of localizing an object based on Bartholomew really was. One was Sam the readings from the detectors, can be nonunique. Elephant and the other was Short Eddy, a This ambiguity is further discussed for a typical fourteen-year-old kid who did not have a whole “source localization” problem, such as lot of normal friends but was superb in math and pinpointing a radioctive source with a set of science. One day an evil villain Hrindar the detectors. It is shown for the planar problem that platypus kidnapped Sam. Bartholomew, as soon the “false source” is the inverse point of the real as he realized Sam was missing, hopped straight one relative to the circle passing through a set of “to Sam the Elephant.” When he got there, he was three detectors. This observation provides an shocked to see Sam chained to a ship anchored in insight leading to an unambiguous pinpointing of the ocean. As soon as Sam saw Bartholomew the source. 1 he knew he was going to be okay. He quickly |SA|:|SB|:|SC|=1:2:3 (the square roots of 1/36, 1/9 and quietly whispered, “Bartholomew, I don’t and 1/4). Eddy always tried to break a complex exactly know where we are, but it is somewhere problem into smaller parts. Therefore, he decided near Landport, Maine.” It was dark out and to focus on the two lighthouses, A and B, first. Bartholomew could hardly see anything but the Apparently, S is one of all possible points P two blurred outline of the city on his left, and the times more distant from B than from A: lights from three lighthouses. Two of them, say A ||/||1/2PA PB = . This observation immediately and B, were on land, while the third one, C, was reminded Eddy of something that had been positioned on the large island. Using the discussed in his AP geometry class. At that time photometer from his spy tool kit, Bartholomew he was very surprised to learn that in addition to found that their brightnesses were in proportion being the locus of points P equally distant from a 36:9: . He hopped to Eddy and told him what was 4 center, a circle can also be defined as a locus of up. Eddy immediately googled the map of the points whose distances to two fixed points A and area surrounding Landport that showed three B are in a constant ratio. Eddy opened his lecture lighthouses (see Fig. 1). ABC turned out to be a notes and... There it was! The notes read: “Circle right triangle, with its legs |AB|=1.5 miles and of Apollonius ... is the locus of points P whose |AC|=2 miles. The accompanying description distances to two fixed points A and B are in a asserted that the lanterns on the lighthouses were constant ratioγ : the same. In a few minutes the friends knew the ||/||PA PB = γ (1) location of the boat, and in another half an hour, For convenience, draw the x-axis through the still under cover of the night, a group of points A and B. It is a good exercise in algebra commandos released Sam and captured the and geometry (see the Appendix) to prove that the villain. The question is, how did the friends radius of this circle is manage to find the position of the boat? ||AB R0 = γ (2) |1|γ 2 − Discussion and solution and its center is at Being the best math and science student in his γ 2 x − x class, Eddy immediately figured out that the ratio x = B A (3) O 2 of the apparent brightnesses could be transformed γ −1 into the ratio of the distances. According to the The examples of the Apollonius circles with the inverse square law, the apparent brightness fixed points A and B corresponding to different (intensity, luminance) of a point light source (a values of γ are shown in Fig. 2. Each of the reasonable approximation when the dimensions of Apollonius circles defined by Eq. 1 is the the source are small compared to the distance r inversion circle [3] for the points A and B from it) is proportional to P / r2 , where P is the (in other words, it divides AB harmonically): 2 power of the source. Given that all lanterns have (x A − xO ) ⋅ (xB − xO ) = RO (4) equal power P, the ratio of the distances between This result immediately follows from Eqs. 2 and the boat (“S” for Sam) and the lighthouses is 3. (Apollonius of Perga [261-190 b.c.e.] was 2 known to contemporaries as “The Great soon Big Sam was released. Once again, the Geometer”. Among his other achievements is the knowledge of physics and math turned out to be famous book Conics where he introduced such very handy. commonly used terms as parabola, ellipse and hyperbola [2])”. 3. The question of ambiguity in some source localization problems Equipped with this information, Eddy was able to draw the Apollonius circle L1 for the points A Our friends have noticed that the solution of their and B, satisfying the condition γ =1/2 (Fig. 3). problem was not unique. The issue was luckily Given |AB|=1.5 and Eq. 2, he found that the radius resolved, however, because the “fictitious” of this circle R1 = 1 mile. Using Eq. 3, he also location happened to be inland. In general, such an ambiguity can cause a problem. Had both the found that xO − x A = −0.5mile which implies intersection points appeared in the ocean, the that the center O of the circle L1 is half a mile to evil villain would have had a 50:50 chance to the south from A. In the same manner Eddy built escape. Thus, it is critical to learn how to deal the Apollonius circle L2 for the points A and C, with this ambiguity in order to pinpoint the real corresponding to the ratio γ =|PA|/|PC|=1/3. Its target and to discard false solutions. radius is R2 = 0.75 mile and the center Q is 0.25 We address this issue using a slightly different mile to the West from A. Eddy put both circles setting, quite typical for the localization on the map. Bartholomew was watching him, and problems. In the previous discussion a measuring holding his breath. “I got it!”- he suddenly tool, the photo detector, was positioned right on shouted. “Sam must be located at the point that the object (the boat) while the physical signals belongs simultaneously to both circles, i.e. right used to pinpoint the boat were produced by the in their intersection. Only in this point his lanterns. More commonly, the signal source is the distance to A will be 2 times smaller than the searched object itself, and the detectors are distance to B and at the same time 3 times smaller located in known positions outside the object. than the distance to C ”. “Exactly!”- responded Practical examples are a radioactive source whose Eddy, and he drew two dots, gray and orange. position must be determined using Geiger Now his friend was confused: “If there are two counters, or a light source detected by the light possible points, how are we supposed to know sensors. Assuming that the source and detectors which one is the boat?” “That’s easy”- Eddy are positioned in the same plane, there are three smiled joyfully- “The gray dot is far inland which unknown parameters in the problem: two leaves us with only one possible location!”. And coordinates, and the intensity of the source P. Eddy drew a large bold “S” right next to the One can suggest that using three detectors should orange dot. Now it was peanuts to discover that be sufficient for finding all the unknowns. The the boat with poor Big Sam was anchored corresponding solution, however, will not be approximately 0.35 miles East and 0.45 miles unique: in addition to the real source, it will North from A.
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