Descartes' theorem
In geometry describes the set of Descartes ( four -circle set of Descartes ), named after René Descartes, a relationship between four circles which touch each other. The kit can be used to find three given circles that touch each other reciprocally a fourth that touches the other three. It is a special case of the Apollonian problem.
- 4.1 Example
History
About geometrical problems associated with circles, which touch each other years was been thinking more than 2000. In ancient Greece the 3rd century BC, Apollonius of Perga about this as a whole book. Unfortunately, we do not get this work, entitled About touches.
René Descartes in 1643 mentioned the problem (according to the former practice ) briefly in a letter to the Princess Elizabeth of Bohemia. He came essentially to the solution, which is described below in equation (1). Therefore, the four- circles set is now named after Descartes.
The British amateur mathematician Philip Beecroft (1842 ) and Frederick Soddy discovered the equation in 1936 again. One speaks sometimes of the Soddy circles, perhaps because Soddy published his version of the sentence in the form of a poem entitled The Kiss Precise, which in the journal Nature (June 20, 1936) was published. Soddy also generalized the theorem of Descartes to a theorem on spheres in n-dimensional space.
Definition of the signed curvature
The set of Descartes is most easily expressed in terms of the notion of curvature. The signed curvature of a circle is defined by, where r is the radius. The larger the circle, the smaller the amount of its curvature, and vice versa.
The minus sign applies in a circle tangent to the other three circles including. Otherwise, the plus sign is to be set.
Consider a straight line as a degenerate circle with curvature so can the set of Descartes also apply when a straight line and are given two circles which touch each other, and a third circle is sought, the affected line of the given circles.
Set of Descartes
Solving this equation for makes it possible to determine the radius of the fourth circle:
The plus - minus symbol expresses that there are two solutions in general.
Example
Given three circles with radii, and. Accordingly, the signed curvature values , and. From equation ( 2) now yield the two solutions and. The tiny circle (red) between the given circles, therefore, has the radius. The large circle ( also red) that includes the given groups, having the radius.
Special cases
For example, the third of the three given circles replaced by a straight line, it is equal to 0, and drops out of equation (1). Equation (2) is much easier in this case:
Example
Given two circles with the radii and and a straight line which is perceived as a circle with infinite radius. The corresponding values for the signed curvature are and. By using Equation (3), two possible values are again obtained, namely, and. For the radii of the two circles drawn in red It therefore follows respectively.
The set of Descartes can not be applied when two or even all three given circles are replaced by straight lines. The set also does not apply if there is more than one enclosing circle tangent, ie in the case of three concentric circles with a common touch point located.
Complex set of Descartes
In order to completely determine a circle, not only its radius (or its curvature ), one must also know its center. The equation for it can be expressed most simply, if one interprets the coordinates of the center point (x, y) as a complex number. The equation for is the set of Descartes very similar and is therefore referred to as a complex set of Descartes.
Given four circles with the centers and the signed curvatures (see above), which touch each other. Then, in addition to (1) the following relationship applies
By substituting yields:
This equation is analogous to and has the solution:
Again, resulting in general two solutions.
Has is determined from equation ( 2), so obtained by