On the area of constrained polygonal linkages

TL;DR

This paper investigates the configuration spaces of polygonal linkages with diagonal constraints, demonstrating that the oriented area functions as a Bott-Morse function and analyzing its critical points and indices.

Contribution

It generalizes previous results on polygonal linkages to include more complex graphs like partial two-trees, providing a detailed analysis of the area function.

Findings

Oriented area is a Bott-Morse function on the configuration space.

Critical points of the area function are characterized.

Bott-Morse indices are computed for these critical points.

Abstract

We study configuration spaces of linkages whose underlying graph are polygons with diagonal constrains, or more general, partial two-trees. We show that (with an appropriate definition) the oriented area is a Bott-Morse function on the configuration space. Its critical points are described and Bott-Morse indices are computed. This paper is a generalization of analogous results for polygonal linkages (obtained earlier by G. Khimshiashvili, G. Panina, and A. Zhukova).

Equations

$$2S(P)=(x_1{y}_2-x_2{y}_1)+\dots +(x_n{y}_1-x_1{y}_n).$$

$$\mu_{P}(S)=\left\{\begin{array}[]{ll}e(P)-1-2\omega_{P}&\hbox{if }\ \sum_{i=1}^{n}\varepsilon_{i}\tan\alpha_{i}>0;\\ e(P)-2-2\omega_{P}&\hbox{otherwise}.\qed\end{array}\right.$$

$${ϵ}_1{l}_1+\cdots +{ϵ}_n{\text{l}}_n=0,\text{mbox}whereϵ\in \{-1,0,1\}$$

$$2S=a_1{a}_2\cdot \sin \alpha +b_1{b}_2\cdot \sin \beta .$$

$$a_1^2+a_2^2-2a_1{a}_2\cdot \cos \alpha =b_1^2+b_2^2-2b_1{b}_2\cdot \cos \beta$$

$$a_1^2+a_2^2-2a_1{a}_2\cdot \cos \alpha =c_1^2+c_2^2-2c_1{c}_2\cdot \cos \gamma .$$

$$M=\left(\begin{array}[]{ccc}a_{1}a_{2}\cdot\cos\alpha&b_{1}b_{2}\cdot\cos\beta&0\\ 2a_{1}a_{2}\cdot\sin\alpha&-2b_{1}b_{2}\cdot\sin\beta&0\\ 2a_{1}a_{2}\cdot\sin\alpha&0&-2c_{1}c_{2}\cdot\sin\gamma\\ \end{array}\right).$$

$$detM=4c_1{c}_2{a}_1{a}_2{b}_1{b}_2\cdot \sin \gamma \cdot \sin (\alpha +\beta ),$$

Full text

This paper is a generalization of analogous results for polygonal linkages (obtained earlier by G. Khimshiashvili, G. Panina, and A. Zhukova).

1. Introduction

A polygonal linkage is a linkage whose underlying graph is a polygon (or, equivalently, a single-cycle graph). One thinks of it as of a flexible polygon with rigid edges and revolving joints at the vertices whose ambient space is the Euclidean plane. The idea of considering the oriented area as a Morse function on its configuration space has already led to some non-trivial results: the critical points (or, equivalently, critical configurations) are easily describable, and there exists a short formula for the Morse index [khipan], [khipan1], [panzh], [zhu]. In some further generalization [PanPerfMorse] the oriented area proves to be an exact Morse function.

In the present paper we extend the class of underlying graphs of linkages in such a way that it is possible to introduce the oriented area with the same nice Morse-theoretical properties.

We start in Section 3 with three-chain linkages. It is our first example which is not a polygonal linkage. By definition, a three-chain linkage is a patch of three chains, and therefore the underlying graph has three cycles. One of the cycles is distinguished: we consider its area $S$ as the function defined on the configuration space. Generically, it is a Bott-Morse function. We prove that the critical configurations are characterized by a combination of* cyclic* and aligned conditions. We also give a formula for Bott-Morse indices of critical points and critical components (Section LABEL:SecMorseXXX).

As an interesting illustration, we describe a Hessian bifurcation and show that it amounts to a pitchfork bifurcation (Section LABEL:SecSmallEx).

Finally, we present the most general result which includes polygonal linkages and three-chains (Section LABEL:SecPTT). Namely, we show that critical points characterization and Bott-Morse index formula extend to partial two-trees. This class of graphs is well-studied and widely used in computer science since many algorithmic problems may be solved much more efficiently for partial k-trees, and in particular for partial two-trees, see [site]. Partial two-trees have a number of graph-theoretical characterizations; we make use of some of them. Also linkages with underlying partial two-tree graph have some specific properties, see [GaoSitharam]; the present paper describes yet another one.

Further generalization of our theory (that is, beyond partial two-trees) needs more complicated criteria of critical points; we present an example.

2. Preliminaries and notation

A * linkage* is a graph together with length assignment to its edges. Its *configuration space * is the space of all planar realizations of the linkage. Its *reduced configuration space * is the quotient of the configuration space by orientation-preserving isometries of the plane, that is, by translations and rotations.

**A realizability convention ** We assume throughout the paper that the edge lengths of a linkage are always realizable, so that the configuration space is non-empty.

A configuration of a linkage is called aligned if it fits in a straight line.

Assume that $X\neq Y$ are two vertices of a configuration $P$ of some linkage. If (1) $P$ is a smooth point of the configuration space, and (2) the gradient of the distance function $|XY|$ does not vanish at $P$, we say that * for the configuration $P$, the vertex $Y$ moves freely with respect to $X$*.

This can be reformulated in two ways: (1) the map from the configuration space to the plane which sends a configuration to the vector $\overrightarrow{XY}$ is a submersion, and (2) the vertex $Y$ admits a non-zero infinitesimal motion in any direction in the plane (with $X$ fixed).

Clearly, this notion is symmetric: for a given configuration, $X$ moves freely with respect to $Y$ iff $Y$ moves freely with respect to $X$. An example follows in Lemma 1.

2.1. Open chains

An open chain is a linkage whose underlying graph is a path graph. In robotics it is also called a robot arm. We assume that its vertices are numbered, so an open chain has a natural orientation. For a configuration $(p_{1},...,p_{n+1})$ of an open chain consider the distance $D=|p_{1}p_{n+1}|$ as a Morse function on the reduced configuration space $(S^{1})^{n-1}$. Its critical points are described in [Farber]:

Theorem 1**.**

[MilKap]

1. (1)

Critical configurations of the distance function $D$ are aligned configurations. 2. (2)

The Morse index of a critical configuration equals $f-1$, where $f$ is the number of forward edges, i.e. the number of (directed) edges $\overrightarrow{p_{i}p}_{i+1}$ which have the same direction as the vector $\overrightarrow{p_{1}p}_{n+1}$.

Let us reformulate the first statement of the theorem in a way to be used in the paper:

Lemma 1**.**

*For any configuration of an open chain, the terminal point

$T=p_{n+1}$ moves freely with respect to the initial point $I=p_{1}$ unless the configuration is aligned. ∎*

2.2. Oriented area for polygonal linkages

A polygonal linkage is a linkage with a single-cycle underlying graph. We assume that all the vertices are numbered, so the cycle carries a natural orientation.

Throughout the paper we assume that no configuration fits a straight line. This is equivalent to smoothness of reduced configuration space of the polygonal linkage, see [Farber]. Let us be more precise. The parameter space $\mathbb{R}^{n}=(l_{1},...,l_{n})$ is divided into *chambers * by hyperplanes (called ”walls”) of type $\sum_{1}^{n}\pm l_{i}=0$ (for all possible combinations of $\pm$’s). If $(l_{1},...,l_{n})$ lies on no wall, the reduced configuration space is smooth. For two $(l_{1},...,l_{n})$ belonging to one and the same chamber the reduced configuration spaces are diffeomorphic.

Definition 1**.**

The oriented area of a polygon with the vertices

$p_{i}=(x_{i},y_{i})$ is defined by

[TABLE]

Definition 2**.**

A polygon $P$ is cyclic if all its vertices $p_{i}$ lie on a circle.

Cyclic polygons arise in the framework of the paper as critical points of the oriented area. The following fact was observed first by Thomas Banchhoff (unpublished):

Theorem 2**.**

[khipan]* Generically, $S$ is a Morse function. At smooth points of the reduced configuration space, a polygon $P$ is a critical point of the oriented area $S$ iff $P$ is a cyclic configuration. ∎*

We recall a short formula for Morse index of a cyclic configuration from [zhu], [khipan1]. We fix the following notation for a cyclic configuration.

$\omega_{P}$ is the winding number of $P$ with respect to the center of the circumscribed circle $O$.

$\alpha_{i}$ is the half of the angle between the vectors $\overrightarrow{Op_{i}}$ and $\overrightarrow{Op_{i+1}}$. The angle is defined to be positive, orientation is not involved.

$\varepsilon_{i}$ is the orientation of the edge $p_{i}p_{i+1}$, that is,

\varepsilon_{i}=\left\{\begin{array}[]{ll}1,&\hbox{if the center Olies to the left of }p_{i}p_{i+1};\\ -1,&\hbox{if the centerO lies to the right of }p_{i}p_{i+1}.\end{array}\right.

$e(P)$ is the number of positive entries in $\varepsilon_{1},...,\varepsilon_{n}$.

$\mu_{P}=\mu_{P}(S)$ is the Morse index of the function $S$ in the point P. That is, $\mu_{P}(S)$ is the number of negative eigenvalues of the Hessian matrix $Hess_{P}(S)$.

Theorem 3**.**

[panzh]**, [khipan1] Generically, for a cyclic configuration $P$ of a polygonal linkage,

[TABLE]

3. Oriented area and its critical points for three-chains

Take three oriented open chains $A,B,$ and $Z$, and glue together their initial points $I$, and also glue together their terminal points $T$. The obtained linkage is a three-chain. If $A$,$B$, and $Z$ have $p$, $q$, and $r$ edges respectively, we say that the three-chain linkage has the type $[p,q;r]$.

The underlying graph has three simple cycles: $AB$, $AZ$, and $BZ$.

Let us first look at the reduced configuration space of a three-chain.

Theorem 4**.**

If none of the three cycles $AB$, $AZ$, and $BZ$ of a three-chain linkage has an aligned configuration, the reduced configuration space of the three-chain is a smooth manifold.

Proof. We mimic the proof of the analogous statement for polygonal linkages, see [Farber]. Let us remove the edge number $r$ from the chain $Z$. We obtain ”a polygon $AB$ with a tail” whose reduced configuration space $M$ is smooth. Next we define the function $D$ on $M$ as the (squared) distance between the end of the tail and the point $T$. The reduced configuration space of the three-chain is the level set of the function $D$: it equals $D^{-1}(l)$, where $l$ is the length of the removed edge. It is a smooth manifold unless $l$ is a critical value of the function $D$. Lemma 1 implies that this happens only if one of the cycles gets aligned. ∎

In this way we get a set of linear equations of type:

[TABLE]

The corresponding arrangement of hyperplanes (”walls”) divides the parameter space into convex chambers. In each chamber the reduced configuration spaces are smooth and mutually diffeomorphic.

In the sequel we will concentrate on three-chains with smooth reduced configuration spaces only, but in case of singularities several statements are also valid in the smooth part.

For a three-chain we distinguish the oriented cycle $\Gamma=AB$ assuming that $A$ goes in the positive direction (and therefore, $B$ goes in the negative direction). In figures we indicate the orientation by an arrow, see Fig. 3.

Definition 3**.**

The *oriented area of a three-chain linkage *is the function $S$ defined on the reduced configuration space as the the oriented area of the cycle $\Gamma$.

A remark on genericity. Throughout the paper we assume that the edge lengths are generic. We do not specify the meaning in advance, instead we keep a right to exclude from ”generic” any finite number of manifolds of codimension at least one. So the generic case remains an open and everywhere dense subset of the parameter space.

Let us first study the simplest case: three-chains of type $[2,2;2]$, or generalized Peaucellier-Lipkin linkage. In this case the reduced configuration space is one dimensional, and therefore is a union of circles.

Proposition 1**.**

For a $[2,2;2]$ three-chain, critical points of $S$ fall into two types (see Figure 1):

1. (1)

Circular type: $\Gamma$ is realized as a cyclic polygon, that is, all the vertices of $\Gamma$ lie on a circle. 2. (2)

Aligned type: $Z$ is aligned.

Proof. In the notation of Figure 2 we have

[TABLE]

We use the Lagrange multipliers method with respect to the equations

[TABLE]

and

[TABLE]

The Lagrange multipliers matrix is

[TABLE]

At critical points its determinant vanishes. The direct computation gives

[TABLE]

which vanishes either $\sin\gamma=0$ (aligned type) or $\sin(\alpha+\beta)=0$ (circular type).∎

Theorem 5**.**

For a three-chain linkage all critical points of the function $S$ fall into two (generically disjoint) types, see Figure 3:

1. (1)

Circular type: $\Gamma$ is realized as a cyclic polygon, that is, all the vertices of $\Gamma$ lie on a circle. For each cyclic configuration of $\Gamma$ we have a critical manifold which equals the reduced configuration space of a flexible polygon obtained from $Z$ by attaching the corresponding diagonal of the cyclic polygon. 2. (2)

Aligned type: $Z$ is aligned, and each of $A$ and $B$ are cyclic. The two circumscribed circles generically do not coincide.

The proof comes from using Proposition 1 sufficiently many times. The ruling idea is to freeze some of the joints of the linkage and to get a small linkage treated in Proposition 1. As an example, if $A$ and $B$ contain two bars each, and $Z$ contains three bars, we freeze one of the joints of $Z$ and use Proposition 1 for the first time, then freeze the other joint of $Z$ and use Proposition 1 for the second time.∎

A genericity remark. According to our genericity convention, we may assume that cyclic polygon $AB$ and simultaneously aligned $Z$ never occur. We may also assume that each of the cyclic polygons that appear in the above theorem is a non-degenerate Morse point (considered as a single polygon).

Example 1**.**

Three-chain [2,2;2].

The reduced configuration space in the smooth case consists of either two or four circles, depending on intervals between minimum and maximal distances between the endpoints of the arms $A,B,$ and $Z$. The global picture is very sensitive for the mutual position of these intervals and gives rise to many different cases. There exist $[2,2;2]$ three-chains with 16 critical points (which is the maximum possible number of critical points). On two components of its reduced configuration space we have only one maximum and one minimum of aligned type. The other components contain four circular and two aligned critical points.