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Steiner’s Alternative: An Introduction to Inversive Geometry

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An Introduction to Inversive Geometry

David Sklar

San Francisco State University

dsklar@sfsu.edu

Bruce Cohen

Lowell High School, SFUSD

bic@cgl.ucsf.edu

http://www.cgl.ucsf.edu/home/bic

Asilomar - December 2005

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

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

Where could we go from here?

Four possible applications

Where can’t we go from here?

The Great Poncelet Theorem

Discovering Steiner’s Alternative

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

Basics of Inversive Geometry

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

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

“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

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

A Step Beyond the Basics

Theorem Two non-intersecting circles C and D can always be

transformed, by an inversion, into two concentric circles

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.

The Power of a Point with Respect to a 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 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 Radical Axis of Two Non-Concentric Circles

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.

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 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.

Non-Intersecting Pencil

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

Where Could We Go from Here?

Higher dimensional inversive geometry:

Four Possibilities

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.

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

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

Geometry II

Where Can’t We Go from Here?

1. M. Berger, Geometry I and Geometry II, Springer-Verlag, New York, 1987

- H.S.M. Coxeter & S.L. Greitzer, Geometry Revisited, The Mathematical
- Association of America, Washington, D.C., 1967

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.

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

5. S.Schwartzman, The Words of Mathematics, The Mathematical

Association of America, Washington, D.C., 1994

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

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.

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