Steiner’s Alternative: An Introduction to Inversive

Geometry

Asilomar - December 2005

Bruce CohenLowell High School, SFUSD

bic@cgl.ucsf.eduhttp://www.cgl.ucsf.edu/home/bic

David SklarSan Francisco State University

dsklar@sfsu.edu

Where could we go from here?

Four possible applications

Where can’t we go from here?

The Great Poncelet Theorem

Basics of Inversive Geometry

Inversion in a circle

Lines go to circles or lines

Circles go to circles or lines

Angles are preserved

A very brief history

Plan

Discovering Steiner’s Alternative

Handout

Statement of the theorem

Sketch of the proof

A step beyond the basics

The Reduction of Two Circles

Concepts in the proof

Power, Radical Axis, Coaxial Pencil, Limit Point

Completing the proof

Part I

Discovering Steiner’s Alternative

1

C

D

2

Steiner’s Alternative (or Steiner’s Porism)

1

1 2

Let and be two circles, with inside , and let be a circle externally

tangent to and internally tangent to . Then construct a chain of circles

, , . . . , , . . . , determined bi

C D C D

C D

1 1

1

y , such that for each let be tangent

to , , , and, for 2 distinct from . Steiner's Alternative says that i

i i

i

C D i

1 1

1

Note: It follows that if for all 1, then for any other initial circle we

will have .i

i

i

1 1 1if for some 1, then for any other initial circle we will have .n nn

1

C

D

7

1

C

D

7

Porism: … a finding of conditions that render an existing theorem indeterminate or capable of many solutions. -- Steven Schwartzman, The Words of Mathematics

A Sketch of the Proof of Steiner’s Alternative

Given two nonintersecting circles there exists a continuous, invertible, “circle preserving” transformation from the “plane” to itself that maps the given non-intersecting circles to concentric circles. Letting T denote such a transformation (a specially chosen “inversion in a circle”) we have

T

1T

T

R

Part II

Basics of Inversive Geometry

Inversion in a Circle

Lines go to Circles or Lines

Circles go to Circles or Lines

Angles are Preserved

Summary: Properties of Inversion

Points inside the circle of inversion go to points outside, points outside go to points inside, points on the circle are fixed and, like reflection, the transformation is self inverse

Inversion preserves the family of circles and lines. Specifically:

Circles that don’t pass through the center of the circle of inversion are mapped to circles that don’t pass through the inversion center (but inversion does not send centers to centers)

Circles that pass through the center of the circle of inversion are mapped to lines that don’t pass through the inversion center

Lines that pass through the center of the circle of inversion are mapped to themselves (although their points are not fixed points)

Lines that don’t pass through the center of the circle of inversion are mapped to circles that pass through the inversion center

Inversion is an angle preserving map, like reflection, the angle between the tangent lines of two intersecting curves is the same as the angle between the tangent lines of their image curves

A Brief History of Inversive Geometry

The idea of inversion is ancient, and was used by Apollonius of Perga about 200 BC.

The invention of Inversive Geometry is usually credited to Jakob Steiner whose work in the 1820’s showed a deep understanding of the subject.

The first explicit description of inversion as a transformation of the punctured plane was presented by Julius Plücker in 1831.

The first comprehensive geometric theory is due to August F. Möbius in 1855.

The first modern synthetic-axiomatic construction of the subject is due to Mario Pieri in 1910.

-- Source: Jim Smith

“Jakob Steiner’s mathematical work was confined to geometry. This he treated synthetically, to the total exclusion of analysis, which he hated, and he is said to have considered it a disgrace to synthetical geometry if equal or higher results were obtained by analytical methods.”

-- Source: Wikipedia

Part III

A Step Beyond the Basics

The Reduction of Two Circles Theorem

The proof is (really) constructive. We will show how to find by a compass and straight-edge construction, from the given circles, two points such that inversion in a circle centered at either point sends the given circles to concentric circles. To help understand why the construction works it’s useful to introduce some interesting, and perhaps unfamiliar, concepts about circles. These concepts are power, radical axis, pencil, and limit point.

and .C D

Theorem Two non-intersecting circles C and D can always be transformed, by an inversion, into two concentric circles

The Power of a Point with Respect to a Circle

2 2

If is a circle of radius and is a point at distance from the

center of then the is

r A d

A d r

power of with respect to

1A

2A

3A

d

d

r

r

B

The power of a point on the circle is zero.

The power of a point A inside of the circle is negative and equal to the negative of the square of the distance from A to the point where the chord perpendicular to the radius through A intersects the circle.

The power of a point A outside of the circle is positive and equal to the square of the distance from A to the point of tangency B.

The locus of points that have the same power with respect to two non-concentric circles is a line perpendicular to their line of centers.

The Radical Axis of Two Non-Concentric Circles

Proof Without loss of generality introduce a coordinate system with the x-axis as the line of centers, the origin at the center of one circle and the center of the other at the point (h, 0).

( , )A x y

2r

2d

1r

1d2 2 2 2

1 1 2 2d r d r 2 2 2 2 2 2

1 2( )x y r x h y r

Let ( , ) be a point that has the same

power with respect to each circle, then

A x y

2 2 22 1( )

2

h r rx

h

a line perpendicular to the line of centers

The locus of points that have the same power with respect to two non-concentric circles is called the Radical Axis of the two circles.

( ,0)h

y

x

Radical Axes Examples

Constructing the Radical Axis of Two Non-intersecting Circles

C

D

E

1L

2L

P

Draw a circle that intersects and

whose center is not on their line of centers.

E C D

Draw the line of centers of circles and . C D

1Draw , the radical axis of circles and . L E C

2Draw , the radical axis of circles and . L E D

1 2 and intersect at a point that has

the same power with respect to each of

the , , and .

L L P

E C D

Since has the same power with respect to and it lies on their radical axis,

so the line through perpendicular to their line of centers is the radical axis

of and .

P C D

P

C D

Pencils of Coaxial Circles

The Pencil of Circles determined by two non-concentric circles C and D is the set of all circles whose centers lie on their line of centers, and such that the radical axis of any pair of circles in the set is the same as the radical axis of C and D.

C

D

Intersecting Pencil

C

D

Non-Intersecting Pencil

Limit Points of Pencils of Non-intersecting Coaxial Circles

C

D

Proof of the Reduction of Two Circles Theorem

and .C D

Theorem Two non-intersecting circles C and D can always be transformed, by an inversion, into two concentric circles

Proof of the Reduction of Two Circles Theorem

and .C D

Theorem Two non-intersecting circles C and D can always be transformed, by an inversion, into two concentric circles

Proof of the Reduction of Two Circles Theorem

and .C D

Theorem Two non-intersecting circles C and D can always be transformed, by an inversion, into two concentric circles

Proof of the Reduction of Two Circles Theorem

and .C D

Theorem Two non-intersecting circles C and D can always be transformed, by an inversion, into two concentric circles

Proof of the Reduction of Two Circles Theorem

and .C D

Theorem Two non-intersecting circles C and D can always be transformed, by an inversion, into two concentric circles

Proof of the Reduction of Two Circles Theorem

and .C D

Theorem Two non-intersecting circles C and D can always be transformed, by an inversion, into two concentric circles

Proof of the Reduction of Two Circles Theorem

and .C D

Theorem Two non-intersecting circles C and D can always be transformed, by an inversion, into two concentric circles

Proof of the Reduction of Two Circles Theorem

and .C D

Theorem Two non-intersecting circles C and D can always be transformed, by an inversion, into two concentric circles

Proof of the Reduction of Two Circles Theorem

and .C D

Theorem Two non-intersecting circles C and D can always be transformed, by an inversion, into two concentric circles

Proof of the Reduction of Two Circles Theorem

and .C D

Theorem Two non-intersecting circles C and D can always be transformed, by an inversion, into two concentric circles

Proof of the Reduction of Two Circles Theorem

and .C D

Theorem Two non-intersecting circles C and D can always be transformed, by an inversion, into two concentric circles

Proof of the Reduction of Two Circles Theorem

and .C D

Theorem Two non-intersecting circles C and D can always be transformed, by an inversion, into two concentric circles

Proof of the Reduction of Two Circles Theorem

and .C D

Theorem Two non-intersecting circles C and D can always be transformed, by an inversion, into two concentric circles

Part IV

Where Could We Go from Here?

A more quantitative development of inversive geometry including the concept of the inversive distance between two circles. This would allow the use of a quick computation to tell whether a Steiner chain is finite.

An application of pencils of nonintersecting circles in the study of the three-sphere

William Thomson (Lord Kelvin) used inversion to compute the effect of a point charge on a nearby conductor consisting of two intersecting planes

Higher dimensional inversive geometry:2

2( )r

T v vv

Four Possibilities

From Marcel Berger’sGeometry II

Part V

Where Can’t We Go from Here?

1 2 1

Let and be two circles, with inside . Construct a sequence of points

, , . . . , , . . . on , such that for each the line segment is tangent

to and (for 2) distinct

i i i

C D D D

P P P D i PP

C i

1 from . "Poncelet's Alternative" says thati iPP

1 1 1if for some 1, then for any other initial point we will have .n nP P n P P P

“Poncelet’s Alternative”: The Great Poncelet Theorem for Circles

Despite the similarity in the statements of the

two theorems, Poncelet's theorem remains

much more difficult to prove than Steiner's.

Bibliography

1. M. Berger, Geometry I and Geometry II, Springer-Verlag, New York, 19872. H.S.M. Coxeter & S.L. Greitzer, Geometry Revisited, The Mathematical Association of America, Washington, D.C., 1967

6. J.T. Smith & E.A. Marchisotto, The Legacy of Mario Pieri in Geometry and Arithmetic, Manuscript (email smith@math.sfsu.edu for access)

3. I. J. Schoenberg, “On Jacobi-Bertrand’s Proof of a Theorem of Poncelet”, in Studies in Pure Mathematics to the Memory of Paul Turán (xxx edition),

Hungarian Academy of Sciences, Budapest, pages 623-627.

5. S.Schwartzman, The Words of Mathematics, The Mathematical Association of America, Washington, D.C., 1994

4. C.S. Ogilvy, Excursions in Geometry, Dover, New York, Dover 1990

The Concentric Case

r

r

R

2

R r

sinR r

r R r

1 sin

1 sin

R

r

The chain will close after one circuit if and

It will close after circuits if and only if , with 3.k n

kn k

only if for some integer 3.nn

C

D

P

A B

r

Cr

r

Dr

CPA r r

DPB r r

C DPA PB r r

The locus of centers of circles tangent to circles C and D is an ellipse with foci at the centers of C and D such that the sum of the distance to the foci is the sum of the radii of C and D.

Warm-up Problem 1 (b)