steiner’s alternative: an introduction to inversive geometry asilomar - december 2005 bruce cohen...
TRANSCRIPT
Steiner’s Alternative: An Introduction to Inversive
Geometry
Asilomar - December 2005
Bruce CohenLowell High School, SFUSD
[email protected]://www.cgl.ucsf.edu/home/bic
David SklarSan Francisco State University
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
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
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
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
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
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
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
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 [email protected] 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