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We have implemented the second case. i.e used the three measured initial circles to find up to 8 Apollonius solution circles that are tangent to the three initial circles. One of these eight circles is the final Apollonius Circle (we use the cluster method to find that circle).
Application
In our algorithm to solve IP based Geolocation we have embedded the Apollonius technique to locate the required target. To get the initial three circles of Apollonius we used the minRTTs from the three landmarks with the lowest minRTTs obtained from pinging the target node. Each of these three minRTTs is then used to estimate the distance = minRTT (ms) * alpha * 100 km to get the radius of the circle. In this way we get the three initial circles of Apollonius which are used by our algorithm to find the solution circles that are tangent to the three initial circles. The resultant solution circles are up to eight in number or less. After getting eight or fewer solution circles we need to find the one final circle that is near our target. We have used the cluster approach to find the final circle and then compared this technique with a manual method to view the performance of our cluster approach. (https://confluence.slac.stanford.edu/display/IEPM/Circle+selection+for+Apollonius+results)
Performance of Apollonius
Like in trilateration, in Apollonius we are using distance = RTT * alpha to get the radius of initial three circles of Apollonius. We can not accurately analyse the efficiency of Apollonius Algorithm using delay to distance mapping. So in order to see the actual performace of Apollonius technique we performed a test. We selected known landmarks and a target (which was also a landmark, but acting as target for other landmarks) then calculated distance (http://www.movabletype.co.uk/scripts/latlong.html) between each landmark to that target and hardcoded this distance to code i.e replaced
e.g distance= 786.00.
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SpreadSheet for Results and Comparison
Reference
http://en.wikipedia.org/wiki/Circles_of_Apolloniushttp://mathworld.wolfram.com/ApolloniusCircle.html
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