Circles of Apollonius

The circle of Apollonius is any of several types of circles associated with Apollonius of Perga, a renowned Greek geometer. Most of these circles are found in planar Euclidean geometry, but analogs have been defined on other surfaces; for example, counterparts on the surface of a sphere can be defined through stereographic projection.

The main uses of this term are fivefold:

A circle is usually defined as the set of points P at a given distance r (the circle's radius) from a given point (the circle's center). However, there are other, equivalent definitions of a circle. Apollonius discovered that a circle could also be defined as the set of points P that have a given ratio of distances k = d₁/d₂ to two given points (labeled A and B in Figure 1). These two points are sometimes called the foci.

The Apollonius pursuit problem is one of finding where a ship leaving from one point A at speed v₁ will intercept another ship leaving a different point B at speed v₂. By assumption, the ships travel in straight lines and the ratio of their speeds is denoted as k = v₁/v₂. At the point they meet, the first ship will have traveled a k-fold longer distance than the second ship. Therefore, the point must lie on a circle as defined by Apollonius, with their starting points as the foci.

The circles defined by the Apollonian pursuit problem for the same two points A and B, but with varying ratios of the two speeds, are disjoint from each other and form a continuous family that cover the entire plane; this family of circles is known as a hyperbolic pencil. Another family of circles, the circles that pass through both A and B, are also called a pencil, or more specifically an elliptic pencil. These two pencils of Apollonian circles intersect each other at right angles and form the basis of the bipolar coordinate system. Within each pencil, any two circles have the same radical axis; the two radical axes of the two pencils are perpendicular, and the centers of the circles from one pencil lie on the radical axis of the other pencil.

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