Steiner chain

A Steiner- chain ( also Steiner circle chain ) is a contiguous sequence of finitely many, mutually contacting circles, each of which also two predetermined, non-intersecting circles in geometry - hereinafter referred to as " output circuits " - touches.

The Steiner chain is named after the Swiss mathematician Jakob Steiner, who defined it in the 19th century and many of their properties discovered and described.

• 2.1 Circle Chain Set
• 2.2 centers and contact points

For the following considerations it is helpful to think of the successive construction of a Steiner chain, starting with a " reset circuit " and ending with a " final circle ". In the usually considered closed Steiner chain and the final circle is tangent again the reset circuit too, in an open, this is not the case. The only condition for the output circuits is that they do not touch or intersect each other. This means that either a smaller circle lies completely within a larger, or that the two channels, without touching each other, are located outside each other.

The figure shows the case of a closed Steiner chain with one inside the other output circuits - called " standard case " in the following.

Variants

Besides the standard case, several variants are possible.

Closed, open and multi -cyclic

Typically, the closed Steiner chain is considered, in which the terminating circuit in contact with the start circuit again. If this is not the case, there is a gap (open Steiner chain) between the starting and terminating circuit; this is closed, overlap with the first and the last channel. If you can continue such a chain in another "round " between the output circuits, so that eventually takes place a connection to the first link in the chain, it is called a multi - cyclic chain. The figure on the far right shows a Steiner- chain (transition from the black- lined circles to the green- completed ) second rotating after a first round and then closes.

• Closed, open and multi - cyclic Steiner chain

Open, overlapping

"Multi- cycle "

The figures above show the sake of simplicity the special case of a Steiner chain in the annulus.

Variants of the circle of contact

Not only can ( " side by side " ), the two output circuits into one another or outside each other, and do not necessarily have all touch each other externally the circles of the chain. This results in terms of contact with stakeholders on the following versions ( the circles of the chain are black, blue and red, the output circuits shown):

• Variants of the circle of contact

Variant 2

Variant 3

Properties

Circle Chain Set

A fundamental statement about the Steiner chain is the Steiner circle chain set (also closing Steiner's theorem ):

This means that each one of these chains may proceed by rotating the chain along the original of the output circuits of the source chain. The animation illustrates this situation.

Centers and contact points

The contact points of the circles of a Steiner chain are always on a circle (gold in the animation ).

The centers of the circles lie on a conic. In the standard case (the two output circuits inside the other ) is the ellipse (green in the animation ), whose foci are the centers of the two output circuits. Incidentally, this is always the case when circles tangent to a given circle inside and a further predetermined circle outside - except for the Steiner- chain even with the Pappus chain, the three-dimensional Soddy - Hexlet and the Apollonian circles.

In the other case ( the output circuits are outside of each other) are the center points on a hyperbola.

Generalizations

• Generalizations

Soddy - Hexlet

A generalization of the Steiner- chain could be to allow the two output circuits to affect each other or to overlap. In the first case a Pappus chain are obtained with an infinite number of chain links.

This is Soddy - Hexlet ( see animation ) extended into the third dimension, 6 -membered Steiner chain - from the output circuits as well as the circles of the chain are each balls. The centers of the six balls of the chain ( these form the Hexlet ) lying on the same ellipse as the center points of the circles of the corresponding Steiner chain. The envelope of the spheres is a Dupin Zyklide, the inversion of a torus. The six balls of Hexlets affect not only the inner ( colored red) and the outer " output sphere", but two additional ( not shown in the animation ) balls that are above or below the plane of Hexlet center points.