Problem of Apollonius - Statement of The Problem

Statement of The Problem

The general statement of Apollonius' problem is to construct one or more circles that are tangent to three given objects in a plane, where an object may be a line, a point or a circle of any size. These objects may be arranged in any way and may cross one another; however, they are usually taken to be distinct, meaning that they do not coincide. Solutions to Apollonius' problem are sometimes called Apollonius circles, although the term is also used for other types of circles associated with Apollonius.

The property of tangency is defined as follows. First, a point, line or circle is assumed to be tangent to itself; hence, if a given circle is already tangent to the other two given objects, it is counted as a solution to Apollonius' problem. Two distinct geometrical objects are said to intersect if they have a point in common. By definition, a point is tangent to a circle or a line if it intersects them, that is, if it lies on them; thus, two distinct points cannot be tangent. If the angle between lines or circles at an intersection point is zero, they are said to be tangent; the intersection point is called a tangent point or a point of tangency. (The word "tangent" derives from the Latin present participle, tangens, meaning "touching".) In practice, two distinct circles are tangent if they intersect at only one point; if they intersect at zero or two points, they are not tangent. The same holds true for a line and a circle. Two distinct lines cannot be tangent in the plane, although two parallel lines can be considered as tangent at a point at infinity in inversive geometry (see below).

The solution circle may be either internally or externally tangent to each of the given circles. An external tangency is one where the two circles bend away from each other at their point of contact; they lie on opposite sides of the tangent line at that point, and they exclude one another. The distance between their centers equals the sum of their radii. By contrast, an internal tangency is one in which the two circles curve in the same way at their point of contact; the two circles lie on the same side of the tangent line, and one circle encloses the other. In this case, the distance between their centers equals the difference of their radii. As an illustration, in Figure 1, the pink solution circle is internally tangent to the medium-sized given black circle on the right, whereas it is externally tangent to the smallest and largest given circles on the left.

Apollonius' problem can also be formulated as the problem of locating one or more points such that the differences of its distances to three given points equal three known values. Consider a solution circle of radius r_s and three given circles of radii r₁, r₂ and r₃. If the solution circle is externally tangent to all three given circles, the distances between the center of the solution circle and the centers of the given circles equal d₁ = r₁ + r_s, d₂ = r₂ + r_s and d₃ = r₃ + r_s, respectively. Therefore, differences in these distances are constants, such as d₁ − d₂ = r₁ − r₂; they depend only on the known radii of the given circles and not on the radius r_s of the solution circle, which cancels out. This second formulation of Apollonius' problem can be generalized to internally tangent solution circles (for which the center-center distance equals the difference of radii), by changing the corresponding differences of distances to sums of distances, so that the solution-circle radius r_s again cancels out. The re-formulation in terms of center-center distances is useful in the solutions below of Adriaan van Roomen and Isaac Newton, and also in hyperbolic positioning or trilateration, which is the task of locating a position from differences in distances to three known points. For example, navigation systems such as LORAN identify a receiver's position from the differences in arrival times of signals from three fixed positions, which correspond to the differences in distances to those transmitters.