Steiner Chain - Elliptical/hyperbolic Locus of Centers

Elliptical/hyperbolic Locus of Centers

The centers of the circles of a Steiner chain lie on a conic section. For example, if the smaller given circle lies within the larger, the centers lie on an ellipse. This is true for any set of circles that are internally tangent to one given circle and externally tangent to the other; such systems of circles appear in the Pappus chain, the problem of Apollonius, and the three-dimensional Soddy's hexlet. Similarly, if some circles of the Steiner chain are externally tangent to both given circles, their centers must lie on a hyperbola, whereas those that are internally tangent to both lie on a different hyperbola.

The circles of the Steiner chain are tangent to two fixed circles, denoted here as α and β, where β is enclosed by α. Let the radii of these two circles be denoted as rα and rβ, respectively, and let their respective centers be the points A and B. Let the radius, diameter and center point of the kth circle of the Steiner chain be denoted as rk, dk and Pk, respectively.

All the centers of the circles in the Steiner chain are located on a common ellipse, for the following reason. The sum of the distances from the kth circle of the Pappus chain to the two centers A and B of thefixed circles equals a constant

\overline{\mathbf{P}_{k}\mathbf{A}} + \overline{\mathbf{P}_{k}\mathbf{B}} = \left( r_{\alpha} + r_{k} \right) + \left( r_{\beta} - r_{k} \right) = r_{\alpha} + r_{\beta}

Thus, for all the centers of the circles of the Steiner chain, the sum of distances to A and B equals the same constant, rα+rβ. This defines an ellipse, whose two foci are the points A and B, the centers of the circles, α and β, that sandwich the Steiner chain of circles. The property of having centers on an ellipse is common to all situations where a series of circles is tangent to two fixed circles, such as the related Pappus chain of circles and the three-dimensional Soddy's hexlet.

The sum of distances to the foci equals twice the semi-major axis a of an ellipse; hence,

2a = r_{\alpha} + r_{\beta}

Let p equal the distance between the foci, A and B. Then, the eccentricity e is defined by 2 ae = p, or

e = \frac{p}{2a} = \frac{p}{r_{\alpha} + r_{\beta}}

From these parameters, the semi-minor axis b and the semi-latus rectum L can be determined

b^{2} = a^{2} \left( 1 - e^{2} \right) = a^{2} - \frac{p^{2}}{4}
L = \frac{b^{2}}{a} = a - \frac{p^{2}}{4a}

Therefore, the ellipse can be described by an equation in terms of its distance d to one focus

d = \frac{L}{1 - e \cos \theta}

where θ is the angle with the line joining the two foci.

Read more about this topic:  Steiner Chain

Famous quotes containing the words locus and/or centers:

    “Seeing the locus of joy as the gate
    of a city, or as a lych-gate ...”
    —Denise Levertov (b. 1923)

    “But look what we have built ... low-income projects that become worse centers of delinquency, vandalism and general social hopelessness than the slums they were supposed to replace.... Cultural centers that are unable to support a good bookstore. Civic centers that are avoided by everyone but bums.... Promenades that go from no place to nowhere and have no promenaders. Expressways that eviscerate great cities. This is not the rebuilding of cities. This is the sacking of cities.”
    —Jane Jacobs (b. 1916)