Planar sections of the quadric of Lie cycles and their ...fillmore/papers/Fillmore...We begin with an overview of those parts of Lie geometry which we employ. Let R 5 be the real five-dimensional

GeometriaeDedicata 55: 175-193, 1995. 175 © 1995 KluwerAcademic Publishers. Printed in the Netherlands.

Planar Sections of the Quadric of Lie Cycles and their Euclidean Interpretations

JAY E FILLMORE 1 and ARTHUR SPRINGER z I Department of Mathematies, University of California at San Diego, LaJolla, CA 92093, U.S.A. ZDepartment of Mathematics, San Diego State University, San Diego, CA 92182, U.S.A.

(Received: 15 September 1993)

Abstract. All cycles (points, oriented circles, and oriented lines of a Euclidean plane) are represented by points of a three dimensional quadric in four dimensional real projective space. The intersection of this quadric with primes and planes are, respectively, two- and one-dimensional systems of cycles. This paper is a careful examination of the interpretation, in terms of systems of cycles in the Euclidean plane, of fundamental incidence configurations involving this quadric in projective space. These interpretations yield new and striking theorems of Euclidean geometry.

Mathematics Subject Classifications (1991): 51 B25, 51M05.

1. Introduction

We start by describing one of the theorems which motivated our investigation.

THEOREM. Consider two circles o f a parabolic Steiner pencil lying on the same side o f the radical axis o f the pencil. Denote by E the unique circle, not in the pencil, whose center is on the line of centers o f the pencil and which is tangent to the two circles. Then: the internal tangential distance from E to any circle tangent to the axis and the larger circle from the pencil, and the external tangentialdistance from E to any circle tangent to the axis and the smaller circle from the pencil, are the same. (See Figure 1.)

This particular theorem has a Euclidean proof which, while not deep, is not trivial. Even with a Euclidean proof of the theorem, what is missing is the guide to its discovery and its correct generalizations. Figure 2 illustrates an apparent generalization. The radical axis is replaced by another circle of the pencil, and E is unchanged. However, x and y are not necessarily equal. A correct generalization is shown in Figure 8 and its reason is discussed in Section 7.

The key to understanding such theorems is Lie's higher geometry of circles in which cycles (points, oriented circles, and oriented lines of a Euclidean plane) are represented by points of a three-dimensional quadric in four-dimensional real projective space. The intersection of this quadric with primes and planes are, respectively, two- and one-dimensional systems of cycles which are called bunches and chains, respectively. This paper is a careful examination of the interpretation, in terms of chains, of some fundamental incidence configurations involving this quadric in projective space.

176 JAY P. FILLMORE AND ARTHUR SPRINGER

Fig. 1. z and g are equal.

) Fig. 2. An apparent generafization, x and y are not necessarily equal.

2. Lie Geometry

Our use of Lie 's higher geometry of (oriented) circles, as a representational geometry,* is in the classical spirit described by Lie [4], Blaschke [1], and Klein [3]. Mos t recently this view finds renewed interest in connection with differential geometry and topology (see Cecil [2]).

The algebraic description of Lie geometry to which this paper is closest is found in Yaglom [6], while the relation to Euclidean theorems is closest in spirit to Rigby [5], both of which appear in the Coxeter Festschrifi.

We begin with an overview of those parts of Lie geometry which we employ. Let R 5 be the real five-dimensional vector space of column vectors t(~o, ~1, ~2,

~ r ~s).** A point of projective space p4 is a one-dimensional subspace (~) of R 5 spanned by a non-zero vector ~.

* The earliest use, of which the authors are aware, of using points of one geometry to represent objects of another is O. Hesse, Uber ein Obertragungspdnzip, J. reine angew. Math. 66 (1866), 15-21.

**'t' denotes transpose. The choice of indices 0, 1,2, r, s, are a mnemonic which will be evident later.

PLANAR SECTIONS OF THE LIE QUADRIC 177

Introduce the inner product

(~1~) = ( i ~ 1 + ~ % 2 _ (r~T _ ~0ns _ ~ s v 0

on R 5 which has signature (+ + + - - ) . This particular 'partial' diagonalization will facilitate geometric interpretations. This inner product determines a three- dimensional quadric

~3 = {(~) E P41 (~l~) --- 0}

in p4, points of which are one-dimensional subspaces (~) spanned by non-zero isotropic vectors ~: (~1~) = 0. If (A) is any point of p4, the three-(projective)- dimensional subspace

(A} ± = {(~} C Pal (A[~) = 0}

is called the polar prime of (A) with respect to the quadric ~2 3. The point (A) is the pole of the prime. The polar of any subspace of p4 is similarly defined.

The quadric f~3 will be called the Lie quadric and its points will be called Lie cycles or just cycles. Cycles will be denoted by lower case Greek letters: (c~), (/3), . . . . Points not on f~3 will be called non-cycles and denoted by upper case Greek letters: (A), (B), . . . .

Two cycles (o~) and (/3) will be said to touch if (c~1/3) = 0. This means that the line (c~, /3) joining the points (c~) and (/3) is a generator of the Lie quadric: (~,/35 c a 3.

Some additional choices are required in order to obtain Euclidean interpretations.

Let Er = t(0, 0, 0, 1, 0)and es = t(0, 0, 0, 0, 1). Then (Er)is anon-cycle and (es} is a cycle. The latter will be called the special cycle. It is the only cycle not admitting a Euclidean interpretation. It plays a role like that of the 'point at infinity' for MObius geometry, and at the same time like that of the special 'spear' in Laguerre geometry.

Points of f~3 Cl (ET) ±, excluding the special cycle (es), will be called point cycles. For a point cycle (~), ff = 0, and then necessarily (0 ¢ 0. We may pass from homogeneous to inhomogeneous coordinates x i = (~/(0 and replace ( by

t ( (x,)2 + (z2)2) = 1, x 1, x 2, 0, 2 "

This point cycle will, of course, be interpreted to be the point of the Euclidean plane having coordinates x ] and x 2.

A cycle (eL} for which a ° ¢ 0 and ~r ¢ 0 will be called a proper cycle. We may write a = t(1, a 1 , a 2, a T, a s) for such a cycle. Note that

(a~ ) 2 + ( a 2 ) 2 - ( a t ) 2 - 2 a s = 0

178 JAY R FILLMORE AND ARTHUR SPRINGER

is a consequence of (o~1o~) = 0. (The other obvious normalization using o~ ~ will be used for 'line cycles' shortly.)

We may confirm that a proper cycle is an oriented circle by finding the locus of point cycles (() which touch a given proper cycle (a). Using a and ( in terms of inhomogeneous coordinates in (o~[() = 0 we have

-2(cq~) = ( x l ) 2 q- ( z 2 ) 2 - 2 a l z 1 - 2a2z 2 + 2a s

= - a ' ) 2 + - a2) _ ( a l ) 2 - ( a 2 7 + 2 a s

= - a ' ) 2 + - a 2 ) 2 _

We recognize this locus as a circle with center at coordinates a 1 and a 2 and radius I a~ I- The sign of the radius a ~ gives an orientation to the circle. The quantity 2a s is the Steiner power of the origin with respect to this circle. We will refer to this circle as 'underlying' the cycle.

Points of ft 3 fq (E s)±, excluding the special cycle (es), will be called a line cycles. For such a cycle (a), o~ ° = 0 and a r ¢ 0, so we may write a = t(0, a l a2 1, a s), where (a 1)2 + (aZ)Z = 1. Like before, we consider point cycles touching (o~) and obtain a line a lz ~ + a2x 2 = a s. The choice of a sign for the unit vector + ( a ~ , a 2) gives an orientation to the line. The quantity laSl is the distance from the origin. We will refer to this line as 'underlying' the cycle.

Note • Two cycles represent the opposite orientations of the same circle if, as points

of p4, they are collinear with the non-cycle (Er). • The prime (es) ± is tangent to the quadric f~3; the intersection f~3 C/(es/± is a

cone with vertex (es). • Two cycles (a) and (/3), each being a proper cycle or a point cycle, are

concentric if the lines (a, r ) and (Er, ~s) in p4 meet. (Thus, the intersection f~3 fq (a, Er, es) is the set of all cycles concentric with (a), together with the special cycle.)

Consider two cycles (a) and (fl), each of which is a proper cycle or a point cycle. By a calculation exactly like that for -2(al~ ), we find

-2(c~lfl) al)2 az)z a~)2. 0~0~ 0 _ _ ( b l _ + ( / ) 2 _ _ ( b r _

We will call this the relative power of (a) and (fl) and denote it by Y( (a ) , (fl)). Relative power may be positive, negative, or zero. When the relative power of

two cycles (a) and (fl), each a proper cycle or a point cycle, is non-negative, it is the square of the length of their common oriented tangents. This will be used systematically later where it will be denoted by Y((a ) , (/3)) = z 2.

Two proper cycles (a) and (fl) touch when (air) = 0 and hence when their relative power is zero. This we recognize as the underlying circles being tangent

PLANAR SECTIONS OF THE LIE QUADRIC

Q'

Fig. 3. Relative power and tangential distance.

179

and having orientations which agree at the point of tangency. Even when relative power is not defined, 'touch' has the expected meaning. For example, two line cycles touch exactly when the underlying lines are parallel and have the same orientation. Also, the special cycle touches exactly line cycles (and itself).

A generator of the Lie quadric is a pencil of cycles which touch. If the generator does not pass through the special cycle, the point, the circles (and line) which underlie the cycles of such a pencil form a parabolic Steiner pencil. If the generator does pass through the special cycle, the lines which underlie the line cycles of the pencil form a pencil of parallel lines.

R e m a r k . Relative power appears in nineteenth-century geometry, but not by that name or use. It is, in fact, a generalization of Steiner power. To see this, consider two oriented circles with centers A and B and unequal (signed) radii a and b, respectively, as in Figure 3. These oriented circles have exactly one oriented center of similitude Z through which there pass (at most) two common (oriented) tangents. If a line through Z meets circle A in P and U , and meets circle B in Q and Q~ with the oriented segment A P parallel or anti-parallel to the oriented segment B Q as the orientation of the two circles agree or disagree, then p Q . p~QI = A B • A B - (a - b) 2 does not depend on the position of the line through Z. This quantity equals the relative power of the two oriented circles.

3. Bunches

A section of the Lie quadric by a non-tangent prime will be called a 'bunch'. Classically these were called linear complexes of cycles, but we prefer to use the name introduced by Yaglom [6].

A point (A) of p4 which is not on the Lie quadric f~3 determines a harmonic homology of t94 which is induced by the reflection in/~5 given by

( ' = ~ - A . 2 (A]() (AIA)"

Reflections, of course, preserve the quadratic form ((1~1), so the harmonic homology sends the quadric f~3 to itself. Acting on the quadric it is called a Lie invers ion. Note


that, as points of projective space p4, the cycle ((), its image (~), and the non-cycle (A) are collinear. The point (A) will be called the pole of the Lie inversion.

The Lie inversion with pole (E~) interchanges the cycles of opposite orientation having the same underlying circle.

Remark. Lie inversions generate a group of transformations called Lie transformations. The connection above, of Lie transformations with reflections in R 5, shows that the group of Lie transformations is isomorphic to the projective real orthogonal group of a quadratic form with signature (+ + + - - ) .

A cycle (~) is fixed under the Lie inversion with pole (A) exactly when (A[~) = 0, so the fixed cycles constitute the points of the Lie quadric lying on the polar prime of the non-cycle (A): f~3 n (A) ±. This set of cycles is called a bunch. The point (A) will also be called the pole of this bunch.

Note that f~3 N (A) ± is a non-singular two-dimensional quadric in the three- dimensional projective space which is the prime (A) ±. A bunch is called positive if this quadric is ruled, and negative if it is not ruled. Positive bunches are exactly those which contain pencils of touching cycles.

Two kinds of cycles associated with a bunch play a central role in our use of Lie geometry.

1. Let f~3 n (A) ± be a bunch. Assume that the line (A, es) meets f~3 in two distinct points. (This requires assuming that (A) does not lie in (es)±. With one exception, this will be satisfied by all bunches discussed in this paper.) One point will necessarily be the special cycle (es); let (e~s) denote the other. The cycle (e~s) will be called the envelope of line cycles of the bunch. This cycle touches every line cycle of the bunch, that is, it touches every cycle in

f~3 N (A) ± n l = ft 3 n <A, e~) ±.

Note that the Lie inversion with pole (A) interchanges the envelope of line cycles and the special cycle.

2. Let f~3 n (A) ± be a bunch. Assume that the line (A, E~) meets ft 3 in two points (#) and (#~), not necessarily distinct. (This will be satisfied by all bunches discussed in this paper.) The cycles (#) and (#') touch every point cycle of the bunch, that is, they touch every cycle in

o (A)" n = a3 n (n ,

The circle (or point) which underlies the two cycles (#) and (#') will be called the locus of point cycles of the bunch. It depends on (A). We will use 'locus of point cycles' for both (#) and (#') without danger of confusion. Again, the notation emphasizes the fact that the Lie inversion with pole (A / interchanges the two cycles (#) and (/z').


It

Fig. 4. A positive bunch as described by Yaglom.

181

Note that when any one of the cycles (#), (#'), (e~) is a proper cycle or a point cycle, the three form a set of concentric cycles. For, (#) and (#~) lie in a 3 n ( 4 , ET,

Figure 6(a) illustrates a Lie inversion.* It should be viewed only as a schematic since the actual three-dimensional (projective) intersection ft 3 n (B) ± lying in a prime (B) -L is frequently a ruled quadric. The location of the pole (B) will be described later.

The key fact relating tangential distances to cycles in a bunch is due to Yaglom [6].

PROPOSITION (Yaglom). Let ft 3 n (A) ± be a bunch whose pole (A) does not lie in the prime (e,) ±, so that the envelope of line cycles (e~) is defined. Then, the relative power Y((¢), (e~s)) of any proper or point cycle (~) of the bunch does not depend on (~). (See Figure 4.)

Proof B y appropriate choice of homogeneous coordinates, we may write

e'~ -- ~ - A . 2 (Ale,) (AIA)"

Note that

-(e'~) ° = (e'~le~) = (e~le~) - (AIe~). 2 (Ale~) - 0 - 2 ( -A0)2 (AIA) (AIA) '

so ° = 2(A°)2/(AIA). If (Q is a proper or point cycle in a 3 N (a} ±, then ~0 ¢ 0 and

* In figures, we will routinely omit the brackets '{ }' designating cycles. This particular figure displays points that will be introduced in the next section.


_ (e~ l~) - (AI,~)2[(AIe~)/(AIA)] 1 ( c t ~0~C0 1 I 0 0

_~-o _ 0 (AIA)

= _12[(A0)2/(AIA)]~0 - (A°) 2"

The left-hand side is the relative power of the cycles (e~s) and (~). []

Remark. The remaining cycles of the bunch are all line cycles which touch (e~).

The application of the proposition of Yaglom to a suitable interpretation of Figure 4 will yield the theorem of Section 1 and its generalizations.

4. C h a i n s

A section of the Lie quadric by a plane will be called a 'chain'. Classically these were called linear congruences of cycles, but we prefer the name used by Yaglom [6]. This use of the term 'chain' is quite different from its classical use in MObius circle geometry.

A (general) chain is the set of cycles of the Lie quadric which lie in a conic in a plane F 2 of p4 :~3 n I ~2. The intersection of this quadric by a plane will, in general, be a conic, possibly degenerate, that is, two lines or a double line. The intersection of the quadric with a plane can also be empty or a double point, but we tacitly exclude this.

Let 9 be the polar line, with respect to the quadric ~2 3, of the plane F 2. Chains are 'classified' by the positions of F 2 and 9 relative to the quaddc ~3 and to the distinguished points (Er) and (es). For example, if 9 meets ft 3 in two distinct points (~1) and (c~2), then the chain ~-~3 n (o~1, O~2) d- consists of all cycles touching the two cycles (oq) and (o~2).

Later (in the lemma of Section 6), we will have occasion to use the following observation: Any non-cycle on 9 is the pole of a bunch containing the chain ~3 n F 2. Except when 9 is a generator of f~3, the chain ~3 n F 2 is contained in a one-parameter family of bunches, and is the intersection of any distinct two of these bunches.

Two kinds of chains, which we call 'cone chains' and 'Steiner chains', will be important for Euclidean interpretations. Although Steiner chains play the lesser role in this paper, we discuss them first since they relate to familiar pencils.

A chain ~3 N F 2 will be called a Steiner chain if the plane F 2 contains the non-cycle (E~).

Two cycles (7) and (7')* which are not the two orientations of the same underlying circle determine the Steiner chain ~'~3 n (~', .~t E~) containing the two cycles.

* For this section we break with the convention that "' denotestheimageofacycleunderspecific Lie inversion. However, when a chain lies in a specific bunch, "' will, in fact, denote the image under the associated Lie inversion.


As with general chains, Steiner chains can be described by the position of the polar line g. For a Steiner chain, this line, as well as the non-ruled quadric f~3 N (E~) z, lie in the three-dimensional projective space (Er) ±. For example, if g meets f~3 N (E~) ± in two distinct points (Trl) and (7r2), then the chain consists of all cycles touching the two point cycles (Trl) and (7r2). The circles underlying these cycles form an elliptic Steiner pencil in the classical sense. In a similar fashion, when g is tangent to the quadric f~3 N (E~) ±, or does not meet it, the chain consists of cycles whose underlying circles form, respectively, parabolic and hyperbolic Steiner pencils in the classical sense.

A chain f~3 N r 2 will be called a cone chain if the plane p2 contains the special cycle (e~).

Two distinct cycles (7} and (7~}, neither of which is the special cycle and which are not a pair of touching line cycles, determine the cone chain f~3 N (7, 7 r, es) containing the two cycles.

Cone chains can be described by the position of the polar line g relative to the cone ~3 N (es)± in the three-dimensional projective space (e~)±, analogous to the description of Steiner chains by the position of 9 relative to the quadric f~3 N (E~) ± in (E~) ±.

Remark. A chain may be both a Steiner chain and a cone chain. 1. If the polar line 9 of F 2 does not pass through (cs), then the plane F 2 =

('7, E~, e~) does not lie in (es} ± and is determined by a single point cycle or proper cycle (7). The chain consists of all cycles concentric with (3'), together with the special cycle.

2. If the polar line g passes through (E~), then the plane p2 = (% E~, zs) lies in (es) ± and is determined by a single line cycle (7). The intersection f~3 N r2 consists of two lines in p4. The chain is determined by a single line and consists of all line cycles whose underlying lines are parallel to this line, together with the special cycle.

Before further describing cone chains, we need to introduce the notions of 'oriented center of similitude' and 'line of centers' for cone chains.

Consider the cone chain f~3 N F 2, If r 2 were to lie in the prime (E~) ±, the cone chain would consist of all point

cycles touching a line cycle (and the line cycle of its opposite orientation). Suppose this is not the case. Then this cone chain contains at most one point cycle:

(~3 n r2) n (E~/± = {(~), ((}},

where (() is a point cycle or is the special cycle. When (() is not the special cycle, it is the (oriented) center of similitude of any two cycles in the cone chain. We then call (() the (oriented) center of similitude of the cone chain. The underlying point is one of the centers of similitude, in the classical sense, of the two circles


underlying any two cycles of the cone chain. The (oriented) center of similitude will appear in the descriptions of cone chains and in figures.

Suppose that the plane F 2 does not contain the non-cycle (E~), so that it is not both a Steiner chain and a cone chain as discussed previously. Then 9 does not lie in (E~) "L and (K) = g n (E~) "L will be a non-cycle that lies in (e~) "L and the cone chain will lie in the bunch f~3 n (K)'L. The locus of point cycles of this bunch is a line which can be shown to be orthogonal to every cycle of the bunch. This line will be called the line of centers of the cone chain. It will appear in figures denoted k.

We assume from now on that the polar line g for a cone chain does not pass through the vertex (es) of the cone.

Since g does not pass through (e,>, there are three kinds of cone chains having both a center of similitude and a line of centers. As these are less familiar and will be central later, we will describe them.

(a) If g meets the cone f~3 n (es)-L in two distinct points (Pl) and (p~), then the chain consists of all cycles touching the two line cycles (Pl> and (P2>. No two cycles of the cone chain touch.

(b) Suppose g is tangent to the cone ~2 3 n (e,> "L at a single point (p>. Then F 2 = ((, p, e~), where (() is the center of similitude. Consequently, f~3 N F 2 = ((, p) U (p, e~) is a singular conic in the plane F 2. It consists of the two lines ((, p> and (p, e~ / meeting at the point (p>. These lines represent, respectively, a pencil which includes proper cycles which touch (() and (p) and a pencil of line cycles which touch (p).

This kind of cone chain will be referred to as 'degenerate', since the conic in F2 is singular.

Alternatively, note that if (7) is a line cycle and (7'> is a point or proper cycle not touching (7), then f~3 fl (7, 7', e~> is a degenerate cone chain. Indeed, (7, e~) is a generator of the quadric f~3 and set (p) = (7, e~) n (7') "L. (This is the same (p> as before.) Then (7, 7', es) contains the pencil (% p) = (7, es) of line cycles as well as the pencil (7', P) consisting of proper cycles (and the point cycle <¢> = <7', p> n

(c) If g does not meet the cone f~3 N <es) -L, no two cycles of the cone chain touch or touch a common line cycle.

These three kinds of cone chains are illustrated in Figure 5(a)-(c). Dashed lines are used to locate the center of similitude in the customary fashion.

5. Determining bunches

The transition between figures in the Euclidean plane and configurations in projective space p4 is facilitated by Figure 6. We continue to view it as a schematic.

In Figure 6(a) we see the pole (A) of a bunch determining the locus of point cycles (#} and (#'}, the envelope of line cycles (e'~>, and the pairs { (7>, (7'> } and


la 2

(a)

I

I ~t P'

I

Co)

(c)

Fig. 5. Cone chains.

. t~Jn < m , ~ J"

(a)

Fig. 6.

~ ~ . ~'"~¢'2>~

(b)

Schematic of configurations in (B) ± and (A) ± to accompany Figure 8.

((6), (6')). Points of ~3 collinear with (A) are images under the Lie inversion with pole (A).


More importantly, in the other direction, we observe that certain configurations of cycles determine bunches. Of several possibilities, we point out a familiar one.

OBSERVATION. Two distinct concentric cycles (each of which is a point cycle or proper cycle) (#) and (~t ) uniquely determine a non-cycle (A) which is the pole o f a bunch whose locus o f point cycles is (#) and whose envelope o f line cycles is

Proof. Since the cycles (#) and (¢~) are concentric, (#, e'~, ET, e~) is a plane and the lines (ET, /z) and (es, e~) meet. Set (A) = (ET, #) R (s~, et~). Then (A) is a non-cycle, since (#), (~ ) , (e~) are distinct and a line not lying in the quadric can meet it in at most two points. The assertions are now clear. []

Remark. This is the case in the Yaglom description of a bunch when I# T/#° I > IT tO lee/s~ [. (See Figure 4.)

6. The Main Theorem

The main theorem is a consequence of:

(1) a lemma which asserts that cycles satisfying certain conditions determine two chains which lie in a common bunch, and

(2) the proposition of Yaglom which tells us that the relative powers (when defined) of cycles in the bunch, with respect to the envelope of line cycles, are the same.

We begin with some generalities regarding two chains. These will, in addition, explain more of Figure 6. Note that Figure 6 illustrates the special case when (7) and (6 / are opposite orientations of the same line or circle. In this case, the four cycles (7), (7'), (~), (~') lie in the same Steiner chain.

OBSERVATION. Let F 2 and A 2 be two distinct planes in p 4 and let g and d be, respectively, their (distinct) polar lines with respect to the quadric Q3. Equivalent are:

(1) The planes r z and A 2 meet in a line. (2) The planes I? 2 and A z lie in a common prime. (3) The lines g and d lie in a plane. (4) The lines 9 and d have a point in common.

The line, prime, plane, and point, determined respectively above, are unique.

CONSEQUENCE. I f two distinct chains f~3 n r 2 and f~3 N A 2 have two distinct cycles in common, they determine the unique bunch f~3 n (F 2, A 2) whose pole is g r id .

APPLICATION. Consider two distinct cone chains

n 3 Iq (7, 7t, ~s) and Q3 n (5, 6.)


which share a common point or proper cycle ( s ) , so that the (distinct) planes

r 2 = (7, 7', ~ ) and A 2 = (6, ~', es)

have the two points (e~) and (e~) in common. These two planes then meet in the line (es, e~s) and lie in a unique prime whose pole we denote by (B):

(r :, (7, 7,, 6, 6', (B) ±.

Necessarily (B) lies in (es)±. The polar lines g = (7, 7') ± N (e~s) ± and d = (~5, c5') ± N (~ )± of these planes meet in the point g N d = (B), and if they meet the quadric meet it in points which are in the line cycles of the chains f~3 N (7, 7') ± and f~3 N (6, 6') ±. (See Figure 6b).

Labels infigures. In Figure 6 and subsequent figures, when applicable, line cycles touching the cycles of the cone chain f/3 n (7, 7', E,) are denoted by (Pl) and (P2): f~3 n 9 = { (Pl), (P2) } (or (p), if there is only one), the center of similitude of this cone chain by ((), the line of centers by k, cycles of the general chain f~3 N (7, 7') ± by (~), and tangential distance from (~) to (() by z. Likewise, for the cone chain f~3 N (g, ~5', es) we use (~rl) and (cr2) for the line cycles, (~) for the center of similitude, g for the line of centers, (~) for cycles of f13 N (6, 6') z, and y for tangential distance. See Figure 7 and 8.

Remark. One can similarly consider two Steiner chains

f ~ 3 N ( 7 , 7 , , E r ) and ~2 3N(6, 6', E~)

having a cycle in common. The pole (B) of their common bunch will then lie in (E~) ±. If the polar lines g and d meet the quadric, they meet it in points which are the point cycles of the chains ~3 n (")', ,./t)Z and f~3 N (6, ~5~) ±.

Further interpretations require careful hypotheses, since planes and lines can have various positions relative to ~3. Some will be the concerns of the next lemmas.

Especially, the observation in Section 5 will be used with cone chains in this section.

Remark. One may also use this observation with Steiner and cone chains to determine Lie inversions from within the Euclidean plane. As this is not central to this paper, it is relegated to an appendix.

LEMMA. Let (7), (7'), (~), (~')be four distinct cycles such that:

(a) (7) and (3") do not touch, and (6) and do not touch

(b) (7), (7'), (~5), (c5') lie in some (general) chain.

Then:


(0) (7, 7') and (6, 6') are distinct coplanar lines of P 4.

Set

<A> = (7, 7'> n <6, 6'>.

Then:

(1) (A) is a non-cycle of P 4. (2) The (general) chains $2 3 O (7, 7'> ± and $2 3 O (8, 6') ± lie in the bunch

a 3 n <A) ±. (3) The Lie inversion with pole (A) interchanges (7} and (7'), and interchanges

<6) and (61).

Proof. (0) By (b), the two lines (7, 7') and (6, 6') lie in a plane. By (a), the line (3', 7') does not lie in the quadric f~3, so it meets f13 in exactly the two points (7) and (7'). Likewise, (8, 6') meets f~3 in exactly (6} and (8'). These four points are distinct, so the two lines are distinct.

(1) The point {A) = {7, 7') N (6, 6') could not lie on f~3 unless one of (7> and (7') coincided with one of (6} and (6').

(2) and (3) are immediate. []

We apply this lemma using (8 / = (e,/, the special cycle, and (6') .= (sls / to obtain:

CONSEQUENCE. Let (7), (7'), (ds) be three arbitrary distinct cycles satisfying:

(a) No one of (7), (7'), (d~) is the special cycle (e~). (b) (7 / and (7') do not touch, and <e~) is a point cycle or proper cycle. (c) (7), (7'), @'s) lie in some cone chain.

Then:

(0) (7, 7') and (e,, e's) are distinct coplanar lines of P 4.

Set

(A> = <-y, -/> n d>.

Then:

(1) (A> is a non-cycle of P 4 not lying in (e,) ±. (2) The (general) chain f~3 O (7, 7') ± lies in the bunch f~3 O {A) ±. (3) The Lie inversion with pole (A} interchanges (7) and (7'). (4) f~3 N (A) ± is a bunch whose envelope of line cycles is ( ds). (5) The relative power Y ( (~), (~'~) ) is the same for every point and proper cycle

(~) in the bunch, especially for every point and proper cycle in the chain Ft 3 n (7, ~/,)_k ofcycles which touch (7) and (7').


i t t

p. 5 "

P2

Fig. 7. Illustrating the main theorem.

Remark. Yaglom [6] uses (2) to define Lie inversion: All cycles of a bunch which touch a cycle (7) also touch a second cycle; the second cycle is (7~).

The main theorem concerns:

• five cycles (7), (7'), (6), (6'), and (e~s), • the cone chains (7, 7', ca) and (6, 6', Es), and • the general chains f~3 N (7, 7') ± and f~3 N (6, 6') ±.

The lemma and its consequence immediately yield the proof.

THEOREM. Let (7), (7'), (6}, (6') be four distinct cycles such that:

(a) (7} and (7') do not touch, and (6) and (6') do not touch. (b) (7), (7'), (6), (6'), lie in some (general) chain. (c) The non-cycle <A> = (7, 7'> N <6, 6'> does not lie in <ca> ±.

Let (c'a) be the envelope o f line cycles of the bunch f~3 n (A) ±. Then: the relative powers with respect to <c'a> of cycles from the two chains ~2 3 N (7, 7'13- and f~3 N (6, 6') -k are the same.

Remark. Note that if one of the four distinct cycles, say (6), is a line cycle, its image (6') under the Lie inversion will touch the envelope of line cycles (eta}. In this case f~3 n (6, 6', es} is a degenerate cone chain. This is the case of the Main Theorem illustrated by Figure 5(b) from the view point of projective space, and by Figure 7 from the viewpoint of the Euclidean plane.

7. The Introduction Revisited

The most striking theorems, such as the one in the introduction, result from the Main Theorem when one cycle of each of the two pairs { (7), (7')} and { (6), (6')},

190 JAY E FILLMORE AND ARTHUR SPRINGER

1

8 f

k

-/

Fig. 8.

%

"t'

Generalization of the theorem of the introduction, x and y are equal.

say (3') and (~), are opposite orientations of the same circle or line. In this case, (7), (7'), (~), (~') lie in the same Steiner chain. Then, as points of p4, (7), (~), and (Er) are collinear, and (7), (7'), (~), (~') are coplanar. In addition, the locus of point cycles (#) (and (#')) then lies in this Steiner chain and is concentric with the envelope of line cycles (e~).

This is illustrated in Figures 6(a) and 8. The latter is the generalization correctly replacing Figure 2. In classical terms, it can be stated as a theorem beginning with three circles or lines in the same (classical) Steiner pencil.

The theorem of Section 1 is the special case that (7) and (6) are two orientations of a line. This forces both of the cone chains (7, 7 ~, es) and (6, 6', e~) to be degenerate. Figure 9 is Figure 1 with appropriate labels.

8. Application

Of several related problems suggested by our colleagues, the most interesting was by D. Wulbert.

Suppose three spheres, whose radii are not necessarily the same, rest on a horizontal plane. Consider all spheres resting on these three. Then the highest points of the latter spheres all lie on a straight line.

Figure 10 is the planar analog. The proof will be clearer if we generalize slightly. Three spheres rest on a plane

G, a sphere G t rests on the three. Let E ~ be a sphere touching G ~ so that the line


5 [ 5'

? tl

Fig. 9. Figure 1 with labels, x and y are equal.

G

Fig. 10. Planar analog of Wulbert 's problem. E ~ is a point.

of centers of E I and G ~ is perpendicular to the plane G. Call one of the initial three spheres X. The plane containing the centers of E I, G t, and X (necessarily orthogonal to the plane G) cuts the spheres in three circles and a line which, when

Et appropriate orientation is chosen, are three proper cycles (~ ) , (7'), (~) and a line cycle (7}. See Figure 11. By the lemma, the relative power of (eta} and (~} does not depend on (~}. Thus the tangential distances from the sphere E t to any of the three original spheres is the same. In case E ~ is a point, it lies on the radical axis of the three spheres.

Appendix: Determining Lie Inversions

It is not easy to describe Lie inversions from within the Euclidean plane. There are two methods. The first uses the Apollonius contact problem. The second uses chains.


I

£s

..~ %,

Fig. 11. Chain associated with Wulbert's problem, e's is a cycle.

USING THE APOLLONIUS CONTACT PROBLEM

To find all cycles which touch three given cycles. Note that, unlike the classical problem, this concerns oriented tangency, so there

are at most two solutions. Let (a l ) , (a2), (~3) be three cycles, no two of which touch. As points of p4, these are three non-collinear points of fZ 3, no two of which lie on a generator. Then

a 3 n ( ( o q > i n ( o ~ 2 ) ± n ( o z 3 ) ± ) = a 3 n ( ( o ~ l , oz2, oz3> ± )

is the set of all cycles that touch each of (0~1) , (oz2) , (oz3). The line (a l , a2, a3) ± meets the quadric f~3 in at most two points.

The Lie inversion with pole (A) fixes every cycle of the bunch ~2 3 N (A)±; any cycle of this bunch touching a cycle (~) also touches its image (~') under the Lie inversion. Thus, to determine the image of a cycle (~) not in the bunch, take any three distinct cycles (a l ) , (a2), (a3) of the bunch which touch (~) (and consequently do not touch each other). The solution of the Apollonius contact problem is the cycle (~) and its image (~').

USING STEINER AND CONE CHAINS

Consider a bunch ft 3 N (A) x for which the locus of point cycles exists, that is, the line (A, E~) meets the Lie quadric (in (#) and (#')). Note first that the pole (A) of a bunch is determined within the Euclidean plane by its locus of point cycles (#)

C t and its envelope of line cycles (e'~). For, (A) = (#, E~) O ( , , es). See Figure 6. A Lie inversion may be described from within the Euclidean plane by the use

of a Steiner chain and a cone chain. Let the cycles (#) and (e'~) be given. To find: The image (( ') of a cycle (~) under the Lie inversion given by the bunch whose locus of point cycles is (#) and whose envelope of line cycles is (E'~).

Now, f~3 N (#, ~, E~ ) is a Steiner chain and f~3 N (e'~, ~, e~) is a cone chain, each containing the cycle (~). These two chains have a second cycle (~') in common and this is the desired image.


Pl

Fig. 12. Lie inversion described using chains in the Euclidean plane.

193

For, the planes (#, (, E~) and (E~s, ~, ~ ) contain the point (~) and the pole (A) of the bunch, so they meet in a line. Then (~') is the cycle correspond- ing to the second intersection of this line with the quadric [23. In fact, the lines (#, Er), (~/s, ~s), (~, ~') are concurrent in (A).

Figure 12 illustrates this method for the chains which we considered in this paper.

References

1. Blaschke, W.: Vorlesungen iiber Differentialgeometrie, III. Differentialgeometrie der Kreise und Kugeln, Springer-Vedag, Berlin, 1929.

2. Cecil, T. E.: Lie Sphere Geometry, Springer-Verlag, New York, 1992. 3. Klein, E: Vorlesungeniiber hrhere Geometrie, 3. Aufl., Springer-Verlag, Berlin, 1926, and Chelsea

Publ. Co., New York, 1957. 4. Lie, S.: Geometrie der Beriihrungstransformationen, Leipzig, 1896, and Chelsea, New York,

1977. 5. Rigby, J. E: The geometry of cycles, and generalized Laguerre inversion, in Davis, Granbaum,

and Sherk (eds), The Geometric Vein, The CoxeterFestschr~, Springer-Verlag, New York, 1981, pp. 355-378.

6. Yaglom, I. M.: On the circular transformations of MObius, Laguerre, and Lie, in Davis, Grfinbaum, and Sherk (eds), The Geometric Vein, The Coxeter Festschrift, Springer-Verlag, New York, 1981, pp. 345-353.

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Planar sections of the quadric of Lie cycles and their ...fillmore/papers/Fillmore...We begin with an overview of those parts of Lie geometry which we employ. Let R 5 be the real five-dimensional