Cross-ratios of quadrilateral linkages

CROSS-RATIOS OF QUADRILATERAL LINKAGES

GIORGI KHIMSHIASHVILI, DIRK SIERSMA

Abstract. We discuss the cross-ratio map of planar quadrilateral linkages,also in the case when one of the links is telescopic. Most of our results arevalid for a planar quadrilateral linkage with generic lengths of the sides.In particular, we describe the image of cross-ratio map for quadrilaterallinkage and planar robot 3-arm.

1. Introduction

We deal with quadrilaterals in Euclidean plane R2 with coordinates (x, y)identified with the complex plane C with coordinate z = x+ ıy. Given such aquadrilateral Q we define cross-ratio of Q as the cross-ratio of the four com-plex numbers representing its vertices in the prescribed order. Using complexnumbers in the study of polygons has a long tradition (see, e.g., [1], [2]). Wepresent several new developments concerned with the above notion of cross-ratio of quadrilateral.

The main aim of this paper is to investigate the values of cross-ratio incertain families of planar quadrilaterals. Two types of such families are dis-cussed: (1) the 1-dimensional moduli spaces of quadrilateral linkage [3] and(2) the 2-dimensional family of a planar robot arm.

In the first part of this paper we deal with quadrilateral linkages (or 4-bar mechanisms [6]). In spite of apparent simplicity of these objects theirstudy is related to several deep results of algebraic geometry and functiontheory, in particular, to the theory of elliptic functions and Poncelet Porism[4]. Comprehensive results on the geometry of planar 4-bar mechanisms arepresented in [6]. Some recent results may be found in [10], [4], [9].

We complement results of [6] and [9] by discussing several new aspects whichemerged in course of our study of extremal problems on moduli spaces of polyg-onal linkages (cf. [8], [9], [10], [11]). In this context it is natural to considerpolygonal linkage as a purely mathematical object defined by a collection ofpositive numbers and investigate its moduli spaces [3]. In this paper we dealwith quadrilateral linkages and planar moduli spaces.

Two types of quadrilateral linkages are considered: (1) conventional quadri-lateral linkages with the fixed lengths of the sides, and (2) quadrilateral linkageswith one telescopic link [3]. Obviously, the latter concept is equivalent to theso-called planar robot 3-arm (or planar triple pendulum [10]). To unify and

Key words and phrases. quadrilateral linkage, robot 3-arm, moduli space, cross-ratio.1

2 GIORGI KHIMSHIASHVILI, DIRK SIERSMA

simplify terminology it is convenient to refer to these two cases by speaking ofclosed and open 4-vertex linkages.

The necessary background for our considerations is presented in Section 2.We begin with recalling the definition and basic geometric properties of planarmoduli spaces of 4-vertex linkages (Proposition 2.1). With a planar 4-vertexlinkage Q one can associate the cross-ratio map CrQ from its planar moduli

space M(Q) into the extended complex plane C (Riemann sphere).Our first main result gives a precise description of the image of cross-ratio

map for a generic quadrilateral linkage (Theorem 3.1). I turns out that cross-ratio is a stable mapping in the sense of singularity theory and that its imageis an arc of a circle or a full circle, depending on the type of the moduli space.This eventually enables us to obtain an analogous result for a planar robot3-arm (Theorem 4.3). Here again cross-ratio is a stable map, having only folds(and no cusps) and the image is an annulus.

In conclusion we mention several possible generalizations of and researchperspectives suggested by our results.

Acknowledgment. Joint research on these topics was started during theauthors’ visit to the Abdus Salam International Centre for Theoretical Physicsin June of 2009. The present paper was completed during a ”Research in Pairs”session in CIRM (Luminy) in January of 2013. The authors acknowledgeexcellent working conditions in both these institutions and useful discussionswith E.Wegert and G.Panina.

2. Moduli spaces of planar 4-vertex linkages

We freely use some notions and constructions from the mathematical theoryof linkages, in particular, the concept of planar moduli space of a polygonallinkage [3]. Recall that closed n-lateral linkage L(l) is defined by a n-tuplel of positive real numbers lj called its sidelengths such that the biggest ofsidelengths does not exceed the sum of remaining ones. The latter conditionguarantees the existence of a n-gon in Euclidean plane R2 with the lengths ofthe sides equal to numbers lj. Each such polygon is called a planar realizationof linkage L(l).

Linkage with a telescopic side is defined similarly but now the last sidelengthln is allowed to take any positive value. For brevity we will distinguish thesetwo cases by speaking of closed and open linkages.

For a closed or open linkage L, its planar configuration space M(L) = M2(L)is defined as the set of its planar realizations (configurations) taken modulothe group of orientation preserving isometries of R2 [3]. It is easy to seethat moduli spaces M(L) have natural structures of compact real algebraicvarieties. For an open n-linkage its planar moduli space is diffeomorphic tothe (n − 2)-dimensional torus T n−2. For a closed n-linkage with a genericside-length vector l, its planar moduli space is a smooth compact (n − 3)-dimensional manifold. As usual, here and below the term ”generic” means

CROSS-RATIOS OF QUADRILATERAL LINKAGES 3

”for an open dense subset of parameter space” (in our setting, this is the spaceRn

+ of sidelengths).In particular, a closed 4-linkage Q = Q(l) is defined by a quadruple of posi-

tive numbers l = (a, b, c, d) ∈ R4+. An open planar 4-linkage (or planar robot

3-arm) is analogously defined by a triple of positive numbers l = (a, b, c) ∈ R3+

and its planar moduli space is diffeomorphic to the two-torus T 2. The completelist of possible topological types of planar moduli spaces of closed 4-linkages isalso well known (see, e.g., [7]).

Proposition 2.1. The complete list of homeomorphism types of planar modulispaces of a 4-bar linkages is as follows: circle, disjoint union of two circles,bouquet of two circles, two circles with two common points, three circles withpairwise intersections equal to one point.

Closed linkages with smooth moduli spaces are called non-degenerate. Itis easy to show that non-degeneracy is a generic condition. In the sequel wemainly focus on non-degenerate quadrilateral linkages, but in our study ofrobot 3-arms we will meet also the degenerate quadrilateral linkages. See thefirst row of Figure 2 for pictures of the moduli spaces.

3. Cross-ratio map of quadrilateral linkage

In this section we use some basic properties of cross-ratio which can be foundin [2]. Recall that the complex cross-ratio of four points (where no three ofthem coincide) p, q, z, w ∈ C is defined as

(1) [p, q; z, w] =z − pz − q

:w − pw − q

=p− zp− w

· q − zq − w

.

and takes values in C∪∞ = P1(C). Coinciding pairs correspond to the values0, 1,∞.Group S4 acts by permuting points so one can obtain up to six values of thecross-ratio for a given unordered quadruple of points which are related by well-known relations [2]. For further use notice also the value of cross-ratio is realif and only if the four points lie on the same circle of straight line [2].

Consider now a quadrilateral linkage Q = Q(a, b, c, d). Note that no threevertices can coincide. Then, for each planar configuration V = (v1, v2, v3, v4) ∈C4 of Q, put

(2) Cr(V ) = Cr((v1, v2, v3, v4)) = [v1, v2; v3, v4] =v3 − v1v3 − v2

:v4 − v1v4 − v2

.

This obviously defines a continuous (in the non-degenerate case actually areal-analytic) mapping CrQ : M(Q) → P1(C). Our main aim in this sectionis to describe its image ΓQ = ImCrQ which is obviously a continuous curvein P1(C). Taking into account some well-known properties of cross-ratio andmoduli space, one immediately obtains a few geometric properties of ΓQ.

In particular, its image should be symmetric with respect to real axis. If thelinkage Q does not have aligned configurations the points of intersection ΓQ


with real axis correspond to cyclic configurations of Q. It is known that Q canhave no more then four distinct cyclic configurations which come into complexconjugate pairs [8]. Hence it ΓQ can intersect the real axis in no more than twopoints. In case M(Q) has two components then they are complex-conjugateand the image of Cr is equal to the image of each component, which impliesthat ΓQ is connected even though M(Q) may have two components.

In further considerations it is technically more convenient to work with an-other map R : M(Q)→ P1(C) defined by the formula

(3) R(v1, v2, v3, v4) = Cr(v1, v3, v2, v4) = [v1, v3; v2, v4] =v2 − v1v2 − v3

:v4 − v1v4 − v3

Figure 1. Quadrilateral.

From the transformation properties of cross-ratio follows that

Cr(V ) = 1−R(V ).

So the properties of Cr can be immediately derived from the properties of R.For brevity we will call R the uniformizer of Q.

The main advantage of R is that, for any configuration V of closed link-age Q, the moduli of numbers vi+1 − vi are constant by its very definition.Consequently, for any V ∈M(Q), one has

(4) |R(V )| = ac

bd.

In other words, R maps M(Q) into the circle of radius acbd

with the center atpoint 0 ∈ C. Later (in the robot arm case) it is more convenient to consider thechart around ∞ and we get a circle with radius bd

ac. Moreover, by elementary

geometric considerations it follows that

(5) argR(V ) = −(α + γ),

where α and γ are the angles at vertices v2 and v4 of configuration V .These observations enable us to get a very precise description of the image

ImR given in the proposition below. Notice that since non-singular modulispace is homeomorphic to a circle or the disjoint union of two circles, one mayuse the natural orientations of M(Q) and 0 ∈ C to define the mapping degreeof uniformizer map.


Theorem 3.1. For a non-degenerate quadrilateral linkage Q, the followingstatements hold:

(1) the image ImR is a subset of the circle of radius ac/bd centered at thepoint 0 ∈ C;

(2) the image ImR is connected and symmetric about the real axis containingthe point ac

bd;

(3) R is surjective if and only if (a+b−c+d)(a−b+c−d)(a−b−c+d) ≤ 0.(4) the mapping degree of R equals zero and multiplicity at each point does

not exceed two.

Proof. The first two statements follow from the preceding discussion. Thethird property can be proved as follows. Take a point eıτ ∈ S1. We wish tosolve the equation ArgR(V ) = τ with V ∈ M(Q). Using the above notationthis is equivalent to solving the system

{a2 + b2 − 2ab cosα = c2 + d2 − 2cd cos γ , α + γ = τ}.Substituting cosα = cos(τ − γ) we get

a2 + b2 − 2ab cos τ cosα− 2ab sin τ sin γ cosα− c2 − d2 + 2cd cos γ = 0.

From this one easily obtains equation of the form

(6) A sin γ +B cos γ = C,

where A = 2ab sin τ, B = 2ab cos τ − 2cd, C = a2 + b2− c2− d2. Now it is easyto see that this equation may have 0, 1 or 2 solutions in [0, 2π] depending onthe sign of expression

Fτ = A2 +B2 − C2 = 4a2b2 + 4c2d2 − (a2 + b2 − c2 − d2)2 − 8abcd cos τ.

Namely, there are no solutions if Fτ > 0, one solution if Fτ = 0, and twosolutions if Fτ < 0.

It is now easy to conclude that if solution exists for certain τ ∈ [0, π] thenit exists for any σ ∈ [0, π], σ < τ because in this case

Fσ(a, b, c, d) ≤ Fτ (a, b, c, d) ≤ 0.

Hence surjectivity takes place if and only if the point with argument π is inthe image. Notice that

Fπ(a, b, c, d) = (a+ b− c+ d)(a− b+ c− d)(a− b− c+ d)(a+ b+ c+ d).

Thus surjectivity is equivalent to Fπ(a, b, c, d) < 0 which differs from the cri-terion of (3) only by a positive factor a+ b+ c+ d. So property (3) is proved.Property (4) follows form the symmetry of R with respect to the real axis,which completes the proof of proposition. �

In the non-degenerate case we have:

Corollary 3.2. The image of Cr is a conjugation-invariant arc of the circleof radius ac/bd centered at the point 1 ∈ C.

This is immediate in view of the relation between Cr and R.


Corollary 3.3. Cross-ratio map of Q is surjective if and only if Q has aself-intersecting cyclic configuration.

This follows from the above proof since the argument of cross-ratio of self-intersecting cyclic configuration is equal to π.

Corollary 3.4. Cross-ratio map of Q is surjective if and only if its planarmoduli space has two components. In other words, surjectivity of cross-ratiomap is a topological property.

Indeed, it was shown in [8] that a self-intersecting cyclic configuration ex-ists if and only if the moduli space has two components. Notice that theseobservations yield a simple criterion of connectedness of the moduli space.

Corollary 3.5. The moduli space is connected if and only if

(a+ b− c+ d)(a− b+ c− d)(a− b− c+ d) ≤ 0.

Notice also that, in non-degenerate case, R(Q) is a smooth mapping be-tween two compact one-dimensional manifolds. One could try to compute itsdifferential with respect to an angular parameter on M2(Q) and identify itscritical points. We do not follow this scenario here because in the next sec-tion we present a more general computation in the case of 3-arm. From thatcomputation follows the next extension of theorem 3.1:

Theorem 3.6. For a non-degenerate quadrilateral linkage Q, the followingstatements hold:

(5) If M(Q) consists of 1 component then cross-ratio is a stable mappingwith exactly 2 fold points. The image is an arc of a circle,

(6) If M(Q) consists of 2 components then cross-ratio is a stable mapping,has no singularities and maps each circle bijective to the image circle.

Figure 2. Moduli spaces of quadrilateral linkages and thecross-ratio images

Next we give the description of the image of R if linkage Q(a, b, c, d) is notgeneric. We will use this in the section about robot arms.


3.1. Long aligned. In this case τ = 0 is the only possibility. We have

F0 = (a− b− c− d)(a+ b+ c− d)(a− b+ c+ d)(a− c+ b+ d) ≥ 0.

The moduli space is a point and the image is just one point.

3.2. Short aligned. If the linkage contains a configurations with τ = π. Con-sequently

Fπ = (a+ b− c− d)(a− b+ c− d)(a− b− c+ d)(a+ c+ b+ d) ≥ 0.

The moduli space is a bouquet of two spheres (at least when no two subsequentedges have the same length) and the uniformizer R maps each of the two circlesonto a full circle. The wedge point is mapped to the intersection of the circlewith the positive real axis.

3.3. Kite. When a = b and c = d we have a moduli space, which consists oftwo circles having two points in common. R maps one circle onto the imagecircle and the other circle collapses to the point on the positive real axis.

3.4. Rhomboid. When a = b = c = d the moduli space consists of threecircles having pairwise a point in common [3]. Note that R maps one circle2 : 1 onto the image circle and the other two circles collaps to the point on thepositive real axis.

4. Cross-ratio map of robot 3-arm

Since the topology of moduli spaces of a robot arm (quadrilateral linkagewith telescopic link) does not depend on the lengths of its bars, we may assumethat the first side is the longest one and a ≥ b ≥ c. Since we are going to dealwith the cross-ratio on M(A), it makes sense to distinguish two cases:

(1) b+ c < a and (2) b+ c ≥ a.

The cross-ratio is defined as a map Cr : M(A) → P1(C) to the Riemannsphere. Only in the second case, M(A(l)) does contain configurations withcoinciding vertices so R attains the value ∞. Since M(A) is diffeomorphicto T 2 we may ask a number of natural questions about the behaviour of Cras a mapping between 2-dimensional manifolds. In particular, in the spiritof Whitney’s results on stable mappings (see, e.g. [2]) one can wonder if Cris stable in the sense of singularity theory: having only folds and cusps assingularities.

We start with a singularity theory approach. Consider the equation

(7) A sinx+B cosx− C = 0

This defines a smooth hyper surface X ⊂ R4. We use coordinates (A,B, x)for X and (A,B,C) for R3. Consider the projection

π : X→ R3,


defined by

π(A,B, x) = (A,B,C) = (A,B,A sinx+B cosx)

The critical set of π is 2-dimensional and smooth, given by A cosx−B sinx = 0.The discriminant (image of the critical set) has equation A2 + B2 − C2 = 0and the image of π is the interior of this cone.

Proposition 4.1. π is a fold map outside the set A = B = 0.

Proof. Follows from a direct computation of the 2-jet of π. The condition for(A0, B0, x0) being a critical point is:

A0 cosx0 −B0 sinx0 = 0.

We write x = x0 +x1, A = A0 +A1 and B = B0 +B1. Modulo constant terms:π(A1, B1, x1) = (A1, B1, C1) where the Taylor expansion (2-jet) of C1 is

(sinx0)A1 + (cosx0)B1 + (A0 cosx0 −B0 sinx0)x1+

+(2 cosx0)A1x1 − (2 sinx0)B1x1 − 12(A0 sinx0 +B0 cosx0)x

21

Modulo contact equivalence we can omit A1- and B1-terms and we get thatC1 is determined by its 2-jet and is equivalent to C0− 1

2C0x

21, which is the fold

map.�

Theorem 4.2. The cross-ratio map for open and closed linkages is a stablemapping with folds only.

Proof. Each linkage determines a map into R3 by putting:

A = 2ab sin τ , B = 2ab cos τ − 2cd , C = a2 + b2 − c2 − d2,depending on τ in the closed quadrilateral case and on (τ, d) in the open case.This is a fold map by transversality. �

Remark: Note that the other type of stable singularity, the cusp, does notoccur!

Next we look for the shape of the image. We slice with circles. Fix a numbert ∈ [0, a + b + c] and consider the 4-bar linkage Qt obtained by adding to Aa fourth side of length t. Then the image of CrA is simply the union of thethe images Cr(M(Qt)). These are arcs of the circles described in Theorem 3.6.

This enables one to decompose the differential of R. We use coordinates(t, α) on M(A) and polar coordinates (|R−1|, argR−1) on the chart at ∞ .Now |R−1(V )| = bt

acand therefore the critical points of R are just the union

of the critical (=fold) points of each of the closed linkages. Level curves areshown in Figures 3 and 4.

Assuming a ≥ b ≥ c there are two generic cases to consider:

i. a > b+ cii. a < b+ c


Figure 3. t-levels;case a > b+ c

Figure 4. t-levels;case a < b+ c

The topology of a slice changes at critical values of t (seen as function M(A)→R). According to [7] these are exactly the aligned positions (where their Morseindices follow from the combinatorics) and in the second case also the value 0.

Figure 5. Slices in case a > b+ c

Figure 6. Slices in case a ≤ b+ c

In case i. we have a Morse function with one maximum, two saddles anda minimum; see the “movie” of cross-ratio slices in figure 5. In case ii. (seefigure 6) there appears an extra saddle and we end up with two minima, whichcorrespond to t = 0 (two conjugate triangles). Pictures of the image of R areshown in figures 7 and 8.

The considerations from section 3 give the following analog of Theorems 3.1and 3.6:


Theorem 4.3. For a generic planar robot 3-arm A(a, b, c), the cross-ratio maphas degree zero, its image is a conjugation-invariant differentiable annulus andbelongs to a disc with radius a + b + c. The cross-ratio map is 2-1 except onthe critical set, with image the fold curves.

Figure 7. Image of R;case a > b+ c

Figure 8. Image of R;case a < b+ c

In order to achieve better understanding of the cross-ratio map in this case,we compute its differential as follows.

We choose a coordinate system on the torus, such that the vertices of thearm are given by 0, a, a+ beiφ, a+ beiφ + ceiη. This gives

R−1 = (a+ beiφ + ceiη)ei(φ−η).

A straightforward computation gives:

Proposition 4.4. The differential of R−1 is given by the formula:

ab sin[φ] + ac sin[η] + bc sin[η − φ]

Corollary 4.5. The Jacobian of R−1 at V ∈ M(A) is equal to the sum oforiented areas of three triangles (see Figure 9)

In the special case of a closed telescopic arm (t = 0) (where polar coordinatesare not well-defined) we have that the Jacobian is equal to bc sin[η − φ]. Thisis non-zero as soon as the arm is not aligned (non-generic case).

Corollary 4.6. The point ∞ ∈ P1(C) is a regular value of uniformizer.

5. Concluding remarks

First of all, we wish to add that using stereographic projection one may in-troduce cross-ratio map for spherical quadrilaterals. The analogs of Theorems3.1 and 4.3 follow in a straightforward way.

It is also interesting to describe the change of cross-ratio under the actionof the so-called Darboux transformation of quadrilateral linkage [4]. Taking


Figure 9. The Jacobian determinant of R

into account a version of Poncelet Porism for quadrilateral linkages obtainedin [4] one might hope to get certain insights concerning the arising discretedynamical system in the image of cross-ratio map.

In a future paper, by a way of analogy we investigate cross-ratios of one-dimensional families of the so-called poristic quadrilaterals arising from Pon-celet Porism [5]. Analogs of our main results are available for bicentric poristicquadrilaterals and poristic quadrilaterals associated with confocal ellipses.

Next, one can also consider cross-ratios of families of quadrilaterals arisingas the centers of circles of Steiner 4-chains [2] and try to describe the image ofthe corresponding cross-ratio map.

Finally, an analogous line of development arises in connection with the no-tion of conformal modulus of a quadrilateral [1]. In particular, one can trydescribe the image and behaviour of conformal modulus for families of poristicbicentric polygons and confocal ellipses. Developments in this direction willbe published elsewhere.

References

[1] L.Ahlfors, Conformal invariants: Topics in Geometric Function Theory, McGraw-Hill,New York, 1973.

[2] M.Berger, Geometrie, Vol.1, Paris, Cedec, 1984.[3] R.Connelly, E.Demaine, Geometry and topology of polygonal linkages, CRC Handbook

of discrete and computational geometry. 2nd. ed., 2004, 197-218.[4] J.Duistermaat, Discrete Integrable Systems, Springer, 2010.[5] L.Flatto, Poncelet’s theorem, Amer. Mat. Soc., Providence, 2009.[6] C.Gibson, P.Newstead, On the geometry of the planar 4-bar mechanism, Acta Appl.

Math. 7, 1986, 113-135.[7] M.Kapovich, J.Millson, On the moduli spaces of polygons in the Euclidean plane, J.

Diff. Geom. 42, No.1, 1995, 133-164.


[8] G.Khimshiashvili, Extremal problems on configuration spaces, Proc. A.RazmadzeMath. Institute 155, 2011, 147-151.

[9] G.Khimshiashvili, Complex geometry of quadrilateral linkages, Proc. Int. Conf. ”Gen-eralized Analytic Functions and their Applications”, 90-100, Tbilisi, 2011.

[10] G.Khimshiashvili, D.Siersma, Cyclic configurations of planar multiple penduli, ICTPPreprint IC/2009/047. 11 p.

[11] G.Khimshiashvili, G.Panina, D.Siersma, A.Zhukova, Critical configurations of planarrobot arms, Centr. Europ. J. Math. 11, 2013, 519-529.

Ilia State University, Tbilisi, Georgia

Utrecht University, The Netherlands

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Cross-ratios of quadrilateral linkages