In geometry, Descartes' theorem, named after René Descartes, establishes a relationship between four kissing, or mutually tangent, circles. The theorem can be used to construct a fourth circle tangent to three given, mutually tangent circles. Table of Geometry, from the 1728 Cyclopaedia. ... René Descartes (March 31, 1596 – February 11, 1650), also known as Renatus Cartesius (latinized form), was a highly influential French philosopher, mathematician, scientist, and writer. ... In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ... Circle illustration This article is about the shape and mathematical concept of circle. ...

History

Geometrical problems involving tangent circles have been pondered for millennia. In ancient Greece of the third century BC, Apollonius of Perga devoted an entire book to the topic. Unfortunately the book, which was called On Tangencies, is not among his surviving works. Apollonius of Perga [Pergaeus] (c. ...

René Descartes touched on the problem briefly in 1643, in a letter to Princess Elizabeth of Bohemia (as such things might go in a letter in those times). He came up with essentially the same solution as given in equation (1) below, and thus attached his name to the theorem. René Descartes (March 31, 1596 – February 11, 1650), also known as Renatus Cartesius (latinized form), was a highly influential French philosopher, mathematician, scientist, and writer. ... Elisabeth, Electress Palatine and (briefly) queen of Bohemia (August 19, 1596 – February 13, 1662), born Princess Elizabeth Stuart of Scotland, was born as the eldest daughter to King James VI of Scotland and his Queen consort Anne of Denmark. ...

Frederick Soddy rediscovered the equation in 1936. The kissing circles in this problem are sometimes known as Soddy circles, perhaps because Soddy chose to publish his version of the theorem in the form of a poem titled The Kiss Precise, which was printed in Nature (June 20, 1936). Soddy also extended the theorem to spheres. Frederick Soddy in 1922. ... Nature is one of the most prominent scientific journals, first published on 4 November 1869. ...

Definition of curvature

Kissing circles. Given three mutually tangent circles (black), what radius can a fourth tangent circle have? There are in general two possible answers (red). The numbers are the circles' curvatures.

Descartes' theorem is most easily stated in terms of the circles' curvature. The curvature of a circle is defined as k = ±1/r, where r is its radius. The larger a circle, the smaller is the magnitude of its curvature, and vice versa. Image File history File links KissingCircles1. ... Curvature refers to a number of loosely related concepts in different areas of geometry. ...

The plus sign in k = ±1/r applies to a circle that is externally tangent to the other circles, like the three black circles in the image. For an internally tangent circle like the big red circle, that circumscribes the other circles, the minus sign applies.

If a straight line is considered a degenerate circle with curvature k = 0, Descartes' theorem also applies to a line and two circles that are all three mutually tangent, giving the radius of a third circle tangent to the other two circles and the line. In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class. ...

Descartes' theorem

If four mutually tangent circles have curvature k_i (for i = 1…4), Descartes' theorem says:

When trying to find the radius of a fourth circle tangent to three given kissing circles, the equation is best rewritten as:

The ± sign reflects the fact that there are in general two solutions. Other criteria may favor one solution over the other in any given problem.

In terms of the circles' radii, the formula is:

Special cases

One of the circles is replaced by a straight line of zero curvature. Descartes' theorem still applies.

If one of the three circles is replaced by a straight line, then one k_i, say k₃, is zero and drops out of equation (1). Equation (2) then becomes much simpler: Image File history File links KissingCircles2. ...

Descartes' theorem does not apply when two or all three circles are replaced by lines. Nor does the theorem apply when more than one circle is internally tangent, e.g. in the case of three nested circles all touching in one point.

Complex Descartes theorem

In order to determine a circle completely, not only its radius (or curvature), but also its center must be known. The relevant equation is expressed most clearly if the coordinates (x, y) are interpreted as a complex number z = x + iy. The equation then looks similar to Descartes' theorem and is therefore called the complex Descartes theorem. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

Given four circles with curvatures k_i and centers z_i (for i = 1…4), the following equality holds in addition to equation (1):

Once k₄ has been found using equation (2), one may proceed to calculate z₄ by rewriting equation (4) to a form similar to equation (2). Again, in general there will be two solutions for z₄, corresponding to the two solutions for k₄.

See also

• Ford circles
• Apollonian gasket
• Soddy's hexlet

In mathematics a Ford circle is a circle with centre at (p/q, 1/2q2) and radius 1/(2q2), where p/q is a fraction in its lowest terms (i. ... In mathematics, an Apollonian gasket or Apollonian net is a fractal generated from three circles, any two of which are tangent to one another. ... Soddys hexlet is a theorem about mutually tangent spheres published by Frederick Soddy in 1937[1], and is the three-dimensional analog of the problem of Apollonius. ...

External links

• Interactive applet demonstrating four mutually tangent circles at cut-the-knot