Skewers

Two lines in 3-dimensional space are skew if they are not coplanar. Two skew lines share a common perpendicular line that we call their skewer. We denote the skewer of lines $a$ and $b$ by $S(a,b)$.¹¹One can also define the skewer of two intersecting lines: it’s the line through the intersection point, perpendicular to both lines.

Consider your favorite configuration theorem of plane projective geometry that involves points and lines. For example, it may be the Pappus theorem, see Fig. 1: if $A_{1},A_{2},A_{3}$ and $B_{1},B_{2},B_{3}$ are two triples of collinear points, then the three intersection points $A_{1}B_{2}\cap A_{2}B_{1}$, $A_{1}B_{3}\cap A_{3}B_{1}$, and $A_{2}B_{3}\cap A_{3}B_{2}$ are also collinear (we refer to [30] for a modern viewpoint on projective geometry).

The Pappus theorem has a skewer analog in which both points and lines are replaced by lines in 3-space and the incidence between a line and a point translates as the intersection of the two respective lines at right angle. The basic 2-dimensional operations of connecting two points by a line or by intersecting two lines at a point translate as taking the skewer of two lines.

In this theorem, we assume that the lines involved are in general position in the following sense: each time one needs to draw a skewer of two lines, this operation is well defined and unique. This assumption holds in a Zariski open subset of the set of the initial lines (in this case, two triples of lines with common skewers, $a_{1},a_{2},a_{3}$ and $b_{1},b_{2},b_{3}$). A similar general position assumption applies to other theorems in this paper.²²The configuration theorems of plane geometry also rely on similar general position assumptions.

Another skewer analog of the Pappus theorem was discovered by R. Schwartz.

Although the formulation of Theorem 2 is similar to that of Theorem 1, we failed to prove it along the lines of the proofs of other results in this paper, and the ‘brute force’ proof of Theorem 2 is postponed until Sect. 6.

Another classical example is the Desargues theorem, see Fig. 2: if the three lines $A_{1}B_{1}$, $A_{2}B_{2}$ and $A_{3}B_{3}$ are concurrent, then the three intersection points $A_{1}A_{2}\cap B_{1}B_{2}$, $A_{1}A_{3}\cap B_{1}B_{3}$, and $A_{2}A_{3}\cap B_{2}B_{3}$ are collinear.

And one has a skewer version:

The projective plane ${\mathbf{RP}}^{2}$ is the projectivization of 3-dimensional vector space $V$. Assume that the projective plane is equipped with a polarity, a projective isomorphism $\varphi:{\mathbf{RP}}^{2}\to({\mathbf{RP}}^{2})^{*}$ induced by a self-adjoint linear isomorphism $V\to V^{*}$.

In particular, in 2-dimensional spherical geometry, polarity is the correspondence between great circles and their poles.³³On $S^{2}$, this is a 1-1 correspondence between oriented great circles and points; in its quotient ${\mathbf{RP}}^{2}$, the elliptic plane, the orientation of lines becomes irrelevant. In terms of 2-dimensional hyperbolic geometry, polarity is depicted in Fig. 3: in the projective model, $H^{2}$ is represented by the interior of a disc in ${\mathbf{RP}}^{2}$, and the polar points of lines lie outside of $H^{2}$, in the de Sitter world.

As a fourth example, consider a theorem that involves polarity, namely, the statement that the altitudes of a (generic) spherical or a hyperbolic triangle are concurrent (in the hyperbolic case, the intersection point may also lie in the de Sitter world).

The altitude of a spherical triangle $ABC$ dropped from vertex $C$ is the great circle through $C$ and the pole $P$ of the line $AB$, see Fig. 4. Likewise, the line $PQ$ in Fig. 3 is orthogonal in $H^{2}$ to the line $AB$.

In the skewer translation, we do not distinguish between polar dual objects, such as the line $AB$ and its pole $P$ in Fig. 4. This yields the following theorem.

In words, the common normals of the opposite sides of a rectangular hexagon have a common normal; see Fig. 5, borrowed from [26].

These ‘skewer’ theorems hold not only in the Euclidean, but also in the elliptic and hyperbolic geometries. In $H^{3}$, two non-coplanar lines have a unique skewer. In elliptic space ${\mathbf{RP}}^{3}$, a pair of generic lines has two skewers; we shall address this subtlety in Sect. 2.3.

In the next section we shall formulate a general correspondence principle, Theorem 5, establishing skewer versions of plane configuration theorems. This correspondence principle will imply the theorems formulated above, except for Theorem 2, whose proof will be given in Sect. 6.

The correspondence principle concerns line geometry of 3-dimensional projective space, a subject that was thoroughly studied in the 19th century by many an eminent mathematician (Cayley, Chasles, Klein, Kummer, Lie, Plücker, Study, $\ldots$) See [20] for a classical and [28] for a modern account.

Although we did not see the formulation of our Theorem 5 in the literature, we believe that classical geometers would not be surprised by it. Similar ideas were expressed earlier. In the last section of [10], Coxeter writes:

$\ldots$ every projective statement in which one conic plays a special role can be translated into a statement about hyperbolic space.

Coxeter illustrated this by the hyperbolic version of the Petersen–Morley theorem.

Earlier Morley [25] also discussed the hyperbolic Petersen–Morley theorem, along with a version of Pascal’s theorem for lines in $H^{3}$ (the “celestial sphere” in the title of this paper is the sphere at infinity of hyperbolic space).

We are witnessing a revival of projective geometry [28]; [30], not least because of the advent of computer-based methods of study, including interactive geometry software (such as Cinderella⁵⁵Which was used to create illustrations in this paper. and GeoGebra). Elementary projective geometry has served as a source of interesting dynamical systems [32]; [33], and it continues to yield surprises [34]. We hope that this paper will contribute to the renewal of interest in this classical area.

Denote the set of lines in 3-space that share a skewer $\ell$ by ${\cal N}_{\ell}$. We saw in Sect. 2 that ${\cal N}_{\ell}$ is an analog of a line in the plane. Two-parameter families of lines in 3-space are called congruences. ${\cal N}_{\ell}$ is a linear congruence: it is the intersection of the Klein quadric with a 3-dimensional subspace ${\mathbf{RP}}^{3}\subset{\mathbf{RP}}^{5}$, that is, it is defined by two linear equations in Plücker coordinates.

Now we describe line analogs of circles.

Let $\ell$ be an oriented line in 3-space (elliptic, Euclidean, or hyperbolic). Let $G_{\ell}$ be the subgroup of the group of orientation preserving isometries that preserve $\ell$. This group is 2-dimensional. Following [29], we call the orbit $G_{\ell}(m)$ of an oriented line $m$ an axial congruence with $\ell$ as axis.

In particular, ${\cal N}_{\ell}$ is an axial congruence.

In $\mathbb{R}^{3}$ (the case considered in [29]), the lines of an axial congruence with axis $\ell$ are at equal distances $d$ from $\ell$ and make equal angles $\varphi$ with it. One defines the dual angle between two oriented lines $\varphi+\varepsilon d$, see [28]. The dual angle between the lines of an axial congruence and its axis is constant.

Thus, in $\mathbb{R}^{3}$, an axial congruence consists of a regulus (one family of ruling of a hyperboloid of one sheet) and its parallel translations along its axis.

Likewise, one defines a complex distance between oriented lines $\ell$ and $m$ in $H^{3}$. Let $d$ be the distance from $\ell$ to $m$ along their skewer $S(\ell,m)$, and let $\varphi$ be the angle between $m$ and the line $\ell^{\prime}$, orthogonal to $S(\ell,m)$ in the plane spanned by $\ell$ and $S(\ell,m)$, and intersecting $m$. (Both $d$ and $\varphi$ have signs determined by a choice of orientation of the skewer). Then the complex distance is given by the formula $\chi(\ell,m)=d+i\varphi$, see [22]. Again, the complex distance between the lines of an axial congruence and its axis is constant.

If $\ell_{1,2}$ and $m_{1,2}$ are the respective points on the sphere at infinity ${\mathbf{CP}}^{1}$ then

where the cross-ratio is given by the formula

see [22].

In the next lemma, ${\mathbf{CP}}^{1}$ is the ‘celestial sphere’, that is, the sphere at infinity of $H^{3}$.

In $S^{3}$, an axial congruence is characterized by the condition that the angles $\alpha$ and $\beta$ (see the proof of Lemma 1) between the axis and the lines of the congruence are constant. It follows from the proof of Lemma 1 that an axial congruence is a torus, a product of circles, one in $S^{2}_{-}$ and another in $S^{2}_{+}$.

Thus an axial congruence of lines is an analog of a circle in 2-dimensional geometry. The arguments from Sect. 2.3 imply analogs of the basic properties of circles:

1. 1.

   If two generic axial congruences share a line then they share a unique other line.

2. 2.

   Three generic oriented lines belong to a unique axial congruence.

(A direct proof of the first property: if the axes of the congruences are $\ell_{1}$ and $\ell_{2}$, and the shared line is $m$, then the second shared line is obtained from $m$ by reflecting in $S(\ell_{1},\ell_{2})$ and reverting the orientation).

Using the approach of Sect. 2, one extends the Correspondence principle to theorems involving circles. For example, one has

As another example, consider the Clifford’s Chain of Circles. This chain of theorems starts with a number of concurrent circles labelled $1,2,3,\ldots,n$. In Fig. 8, $n=5$, and the initial circles are represented by straight lines (so that their common point is at infinity).⁶⁶As usual, lines are considered as circles of infinite radius. The intersection point of circles $i$ and $j$ is labelled $ij$. The circle through points $ij,jk$ and $ki$ is labelled $ijk$.

The first statement of the theorem is that the circles $ijk,jkl,kli$ and $lij$ share a point; this point is labelled $ijkl$. The next statement is that the points $ijkl,jklm,klmi,lmij$ and $mijk$ are cocyclic; this circle is labelled $ijklm$. And so on, with the claims of being concurrent and cocyclic alternating; see [9]; [26], and [21]; [31] for a relation with completely integrable systems.

A version of this theorem for lines in $\mathbb{R}^{3}$ is due to Richmond [29]. The approach of Sect. 2 provides an extension to the elliptic and hyperbolic geometries.

Next, we present an analog of the Poncelet Porism, see, e.g., [12]; [14]. This theorem states that if there exists an $n$-gon inscribed into a conic and circumscribed about a nested conic then every point of the outer conic is a vertex of such an $n$-gon, see Fig. 9.

Consider a particular case when both conics are circles (a pair of nested conics can be sent to a pair of circles by a projective transformation). The translation to the language of lines in space is as follows.

Consider two generic axial congruences ${\cal C}_{1}$ and ${\cal C}_{2}$, and assume that there exist a pair of lines $\ell_{1}\in{\cal C}_{1}$ and $\ell_{2}\in{\cal C}_{2}$ that intersect at right angle. That is, ${\cal C}_{1}$ and ${\cal N}_{\ell_{2}}$ share the line $\ell_{1}$. By property (1) above, there exists a unique other line $\ell_{1}^{\prime}\in{\cal C}_{1}$, shared with ${\cal N}_{\ell_{2}}$, that is, $\ell_{1}^{\prime}$ intersects $\ell_{2}$ at right angle. Then there exists a unique other line $\ell_{2}^{\prime}\in{\cal C}_{2}$ that intersects $\ell_{1}^{\prime}$ at right angle, etc. We obtain a chain of intersecting orthogonal lines, alternating between the two axial congruences.

The following theorem holds in the three classical geometries.

A pair of nested circles in the Euclidean plane is characterized by three numbers: their radii, $r<R$, and the distance between the centers, $d$. The conditions for the existence of an $n$-gons inscribed into one and circumscribed about another circle (a bicentric $n$-gon) are known as the Fuss relations. The first ones, for $n=3$ and $n=4$, are

the case $n=3$ is due to Euler; Fuss found the relations for $n=4,\ldots,8$. More generally, Cayley gave conditions for Poncelet polygons to close up after $n$ steps for a pair of conics, see [12]; [14]).

It would be interesting to find an analog of the Fuss and Cayley relations; up to isometry, a pair of axial congruence depends on 6 parameters (two characterizing each congruence and two describing the mutual position of the axes).

In this section, we propose a definition-construction of a skewer analog of a conic.

Let us first describe a skewer analog of a projection of a line to a line. Figure 10 depicts the central projection $\varphi_{O}:a\to b$ between two lines in the plane.

Consider three lines in space, $a,b$ and $O$, and define a map $\varphi_{O}:{\cal N}_{a}\to{\cal N}_{b}$ as follows: for $\ell\in{\cal N}_{a},$ set $\varphi_{O}(\ell)=S(S(\ell,O),b)$. This is a skewer analog of the central projection. Like in the plane, this operation is involutive: swapping the roles of $a$ and $b$, and applying it to the line $S(S(\ell,O),b)$, takes one back to line $\ell$.

Following Sect. 2.4, one can describe the hyperbolic case of this construction in ${\mathbf{CP}}^{2}$; the result is (a complex version of) the central projection in Fig. 10.

Recall the Braikenridge–Maclaurin construction of a conic depicted in Fig. 11; see [23] for the history of this result.

Fix two lines, $p$ and $q$, and three points, $O,A$ and $B$. Identify $p$ with the pencil of lines through point $A$, and $q$ with the pencil of lines through point $B$. The central projection $\varphi_{O}:p\to q$ induces a projective transformation between the two pencils of lines. Then the locus of intersection points of the corresponding lines from these pencils is a conic.

One can use the skewer version of the Braikenridge–Maclaurin construction to define a line analog of a conic. Start with five lines $O,p,q,A,B$. For each line $\ell\in{\cal N}_{p}$, we have the corresponding line $m=S(S(\ell,O),q)\in{\cal N}_{q}$. Then the 2-parameter family of lines

is a skewer analog of a conic. In the hyperbolic case, this set is identified with a conic in ${\mathbf{CP}}^{2}$.

Given a finite set $S$ of points in the plane, assume that the line through every pair of points in $S$ contains at least one other point of $S$. J.J.Sylvester asked in 1893 whether $S$ necessarily consists of collinear points. See [5] for the history of this problem and its generalizations.

In $\mathbb{R}^{2}$, the Sylvester question has an affirmative answer (the Sylvester–Galai theorem), but in ${\mathbb{C}}^{2}$ one has a counter-example: the 9 inflection points of a cubic curve (of which at most three can be real, according to a theorem of Klein), connected by 12 lines.

Note that the dual Sylvester–Galai theorem holds as well: if a finite collection of pairwise non-parallel lines in $\mathbb{R}^{2}$ has the property that through the intersection point of any two lines there passes at least one other line, then all the lines are concurrent.

The skewer version of the Sylvester Problem concerns a finite collection $S$ of pairwise skew lines in space such that the skewer of any pair intersects at least one other line at right angle. We say that $S$ has the skewer Sylvester property. The question is whether a collection of lines with the skewer Sylvester property necessarily consists of lines that share a skewer.

In this section we prove Theorem 2. This computational proof is joint with R. Schwartz.

As before, it suffices to establish the hyperbolic version of Theorem 2. We use the approach to 3-dimensional hyperbolic geometry, in the upper half-space model, developed by Fenchel [13]; see also [19]; [22]. The relevant features of this theory are as follows.

To a line $\ell$ in $H^{3}$, one assigns the reflection in this line, an orientation preserving isometry of the hyperbolic space, an element of the group $PGL(2,{\mathbb{C}})$. One can lift it to a matrix ${M_{\ell}}\in GL(2,{\mathbb{C}})$, defined up to a complex scalar. Since reflection is an involution, one has ${\rm Tr}(M_{\ell})=0$. More generally, a traceless matrix $M\in GL(2,{\mathbb{C}})$ is called a line matrix; it satisfies $M^{2}=-\det(M)E$ where $E$ is the identity matrix.

The skewer relations translate to the language of matrices as follows:

• •

  two lines $\ell$ and $n$ intersect at right angle if and only if ${\rm Tr}(M_{\ell}M_{n})=0$;

• •

  the skewer of two lines $\ell$ and $n$ corresponds to the commutator $[M_{\ell},M_{n}]$;

• •

  three lines $\ell,m,n$ share a skewer if and only if the matrices $M_{\ell},M_{m}$, and $M_{n}$ are linearly dependent.

Likewise, one assigns matrices to points. The reflection in a point $P$ is an orientation-reversing isometry of $H^{3}$; one assigns to it a matrix $N_{P}$ in $GL(2,{\mathbb{C}})$, defined up to a real scalar, with $\det N_{P}>0$ and satisfying $N_{P}{\overline{N}_{P}}=-\det(N_{P})E$, where bar means the entry-wise complex conjugation of a matrix. Such matrices are called point matrices.

Equivalently, point matrices $N$ satisfy $n_{22}=-{\bar{n}_{11}},n_{12}\in\mathbb{R},n_{21}\in\mathbb{R}$, that is, the real part of $N$ is a traceless matrix, and the imaginary part is a scalar matrix. It is convenient to normalize so that the imaginary part is $E$, and then $N$ can be though of as a real 3-vector consisting of three entries of the real part of $N$.

Incidence properties translate as follows:

• •

  a point $P$ lies on a line $\ell$ if and only of $M_{\ell}N_{P}=N_{P}{\overline{M}_{\ell}}$;

• •

  three points are collinear if and only if the respective point matrices are linearly dependent (equivalently, over $\mathbb{R}$ or ${\mathbb{C}}$).

We need a formula for a line matrix corresponding to the line through two given points. Let $N_{1}$ and $N_{2}$ be point matrices corresponding to the given points. Then the desired line matrix $M\in GL(2,{\mathbb{C}})$ satisfies the system of linear equations

This system is easily solved and it defines $M$ up to a factor (we do not reproduce the explicit formulas here).

With these preliminaries, the proof proceeds in the following steps.

1. 1.

   Start with two triples of linearly dependent point matrices, corresponding to the triples of points $A_{1},A_{2},A_{3}$ and $B_{1},B_{2},B_{3}$.

2. 2.

   Compute the line matrices, corresponding to the lines $(A_{1}B_{2})$ and $(A_{2}B_{1})$, $(A_{2}B_{3})$ and $(A_{3}B_{2})$, and $(A_{3}B_{1})$ and $(A_{1}B_{3})$ by solving the respective systems (2).

3. 3.

   Compute the commutators of these three pairs of line matrices.

4. 4.

   Check that the obtained three matrices are linearly dependent.

We did these computations in Mathematica. Since a line matrix is traceless, it can be viewed as a complex 3-vector, and the last step consists in computing the determinant made by three 3-vectors. The result of this last computation was zero (for arbitrary initial point matrices) which proves the theorem.