Pappus chain

A Pappus chain

In geometry, the Pappus chain is a ring of circles between two tangent circles investigated by Pappus of Alexandria in the 3rd century AD.

Construction

Given two circles Template:Mvar and Template:Mvar, let the inner circle Template:Mvar be enclosed by the outer circle Template:Mvar, and let the two circles be tangent to each other at point Template:Mvar. Let the radii of these two circles be denoted as Template:Mvar, respectively, and let their respective centers be the points Template:Mvar. The Pappus chain consists of the circles in the shaded grey region, which are externally tangent to Template:Mvar (the inner circle) and internally tangent to Template:Mvar (the outer circle). Let the radius, diameter and center point of the Template:Mvar^th circle of the Pappus chain be denoted as Template:Mvar, respectively.

The Pappus chain is often considered with respect to an arbelos, a circular triangle whose three sides are semicircles of the two given tangent circles and of the circle in the chain whose center is collinear with the two given circles.

Properties

Centers of the circles

Ellipse

All the centers of the circles in the Pappus chain are located on a common ellipse, for the following reason. The sum of the distances from the Template:Mvar^th circle of the Pappus chain to the two centers Template:Mvar of the arbelos circles equals a constant

$$\overline{P_{n}U}+\overline{P_{n}V}=(r_U+r_n)+(r_V-r_n)=r_U+r_V$$

Thus, the foci of this ellipse are Template:Mvar, the centers of the two circles that define the arbelos; these points correspond to the midpoints of the line segments Template:Mvar, respectively.

If $r=\frac{\overline{AC}}{\overline{AB}},$ then the center of the Template:Mvarth circle in the chain is:

$$(x_n,y_n)=(\frac{r(1+r)}{2[n^2(1-r{)}^2+r]},\frac{nr(1-r)}{n^2(1-r{)}^2+r})$$

Radii of the circles

If $r=\frac{\overline{AC}}{\overline{AB}},$ then the radius of the Template:Mvarth circle in the chain is: $$r_n=\frac{(1-r)r}{2[n^2(1-r{)}^2+r]}$$

Circle inversion

Under a particular inversion centered on Template:Mvar, the four initial circles of the Pappus chain are transformed into a stack of four equally sized circles, sandwiched between two parallel lines. This accounts for the height formula Template:Math and the fact that the original points of tangency lie on a common circle.

The height Template:Mvar of the center of the Template:Mvar^th circle above the base diameter Template:Mvar equals Template:Mvar times Template:Mvar. This may be shown by inverting in a circle centered on the tangent point Template:Mvar. The circle of inversion is chosen to intersect the Template:Mvar^th circle perpendicularly, so that the Template:Mvar^th circle is transformed into itself. The two arbelos circles, Template:Mvar and Template:Mvar, are transformed into parallel lines tangent to and sandwiching the Template:Mvar^th circle; hence, the other circles of the Pappus chain are transformed into similarly sandwiched circles of the same diameter. The initial circle Template:Math and the final circle Template:Mvar each contribute Template:Math to the height Template:Mvar, whereas the circles Template:Math to Template:Math each contribute Template:Mvar. Adding these contributions together yields the equation Template:Math.

The same inversion can be used to show that the points where the circles of the Pappus chain are tangent to one another lie on a common circle. As noted above, the inversion centered at point Template:Mvar transforms the arbelos circles Template:Mvar into two parallel lines, and the circles of the Pappus chain into a stack of equally sized circles sandwiched between the two parallel lines. Hence, the points of tangency between the transformed circles lie on a line midway between the two parallel lines. Undoing the inversion in the circle, this line of tangent points is transformed back into a circle.

Steiner chain

In these properties of having centers on an ellipse and tangencies on a circle, the Pappus chain is analogous to the Steiner chain, in which finitely many circles are tangent to two circles.

Bibliography

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External links

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