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The Configuration Space of n-Tuples of Equiangular Unit Vectors for n = 3, 4, and 5.

1. Introduction

Recently, starting in [1], the topology of the configuration space of spatial polygons of arbitrary edge lengths has been considered by many authors. In the equilateral case, the definition is given as follows. For l > 0, we set

[P.sub.n](l) = {([a.sub.1], ... [a.sub.n]) [member of] [([S.sup.2]).sup.n]} | [[summation].sup.n.sub.i=1[]la.sub.i] = 0}/SO(3). (1)

Here [a.sub.i] [member of] [S.sup.2] denote the unit vectors in the directions of the edges of a polygon; the group SO(3) acts diagonally on ([a.sub.1], ..., [a.sub.n]).

Many topological properties of [P.sub.n](l) are already known: First, it is clear that there is a homeomorphism

[P.sub.n](l) [congruent to] [P.sub.n](1) we. (2)

Second, it is proved in [2] that [P.sub.5](1) is homeomorphic to del Pezzo surface of degree 5.

Third, when n is odd, the integral cohomology ring [H.sup.*]([P.sub.n](1); Z) was determined in [3]. We refer to [4] for other properties of [P.sub.n](1), which is an excellent survey of linkages.

In another direction, we consider the space of n-tuples of equiangular unit vectors in [R.sup.3]. More precisely, we define the following: We fix [theta] [member of] [0, [pi]] and set

[mathematical expression not reproducible], (3)

where <,> denotes the standard inner product on [R.sup.3]. Using (3), we define

[M.sub.n]([theta]) = [A.sub.n]([theta])/SO(3). (4)

It is expected that the space [M.sub.n]([theta]) is much more difficult than [P.sub.n](l). For example, the following trivial observation shows that [M.sub.n]([theta]) does not admit a similar property to (2): when n is odd, we have [M.sub.n]([theta]) = {one point} but [M.sub.n]([theta]) = [empty set].

We claim that [M.sub.n]([theta]) is a hypersurface of the torus [T.sup.n-2]. In fact, if we forget the condition <[a.sub.n], [a.sub.1]> = cos [theta] in (3), the space corresponding to (4) is [T.sup.n-2] as observed in [5,6]. Hence the claim follows.

We recall previous results on [M.sub.n]([theta]). First, [7] considered the case for [theta] = [pi]/2. The main result is that, realizing [M.sub.n]([pi]/ 2) as a homotopy colimit of a diagram involving [M.sub.n-2]([pi]/2) and [M.sub.n-1]([pi]/2), we inductively computed [chi]([M.sub.n]([pi]/2)). In particular, we obtained a homeomorphism [M.sub.5]([pi]/2) [congruent to] [[summation].sub.5], where [[summation].sub.5] denotes a connected closed orientable surface of genus 5.

Second, we set

[X.sub.n]([theta]) := [P.sub.n](1) [intersection] [M.sub.n]([theta]). (5)

Note that [X.sub.n]([theta]) is the configuration space of equilateral and equiangular n-gons. Crippen [8] studied the topological type of [X.sub.n]([theta]) for n=3,4, and 5. The result is that [X.sub.n]([theta]) is either 0, one point, or two points depending on 0. Later, O'Hara [9] studied the topological type of [X.sub.6]([theta]). The result is that [X.sub.6]([theta]) is disjoint union of a certain number of [S.sup.1]'s and points.

The purpose of this paper is to determine the topological type of [M.sub.n]([theta]) for n =3,4, and 5. In contrast to the fact that at most one-dimensional spaces appear in the results of [8, 9], surfaces appear in our results.

This paper is organized as follows. In Section 2, we state our main results and in Section 3 we prove them.

2. Main Results

Theorem A. The topological type of [M.sub.3]([theta]) is given in Table 1.

Theorem B. (i) The topological type of [M.sub.4]([theta]) is given in Table 2.

(ii) As [theta] approaches [pi]/2, point A in Figure 1(a) approaches point B.

Theorem C. (i) The topological type of [M.sub.5]([theta]) is given in Table 3. Let be a connected closed orientable surface of genus g.

(ii) (a) Let [theta] satisfy that 2[pi]/5 < [theta] < 2[pi]/3. We study the situation where [theta] approaches 2[pi]/3. We identify the torus [[summation].sub.1] with the Dupin cyclide, which we denote by D. (See Figure 2.)

Using this, we identify [[summation].sub.5] with [#.sub.5]D, where the connected sum is formed by cutting a small circular hole away from the narrow part of D. As 9 approaches 2[pi]/3, the center of each narrow part pinches to a point. Thus the five singular points appear.

(b) We consider the situation where 9 increases from 2[pi]/3. Then each pinched point of M5(2[pi]/3) separates. Thus we obtain [S.sup.2].

(c) Let 9 satisfy that 2[pi]/5 < d < 2[pi]/3. We consider the situation where 9 approaches 2[pi]/5. In contrast to (a), the center of exactly one narrow part pinches to a point. Thus one singular point appears.

Corollary D. As a subspace of ([([S.sup.2]).sup.5]x[0, [pi]])/SO(3), wedefine the space

[mathematical expression not reproducible]. (6)

Then [M.sub.5](0) is a singular point of Y and has a neighborhood C[[summation].sub.4], where C denotes the cone.

Remark 1. Cone-type singularities appear in Theorems B and C and Corollary D. We note that singularities of configuration spaces of mechanical linkages have been studied extensively by Blanc and Shvalb [10].

3. Proofs of the Main Results

We fix [theta] [member of] [0, [pi]] and set

[mathematical expression not reproducible]. (7)

Normalizing [a.sub.1] and [a.sub.2] to be [e.sub.1] and p, respectively, we have the following description:

[mathematical expression not reproducible]. (8)

Hereafter we use (8).

In order to prove our main results, we use the following fact, whose proof is left to the reader.

Fact 2. Let ([alpha], [beta], [gamma]) [member of] [([S.sup.2]).sup.3] satisfy that {a, [beta]) = 0,

[mathematical expression not reproducible]. (9)

Then, there exists [phi] [member of] R such that

[gamma] = (cos [theta])[alpha] + (sin [theta] cos [phi]) [beta] + (sin [theta] sin [phi]) ([alpha] x [beta]). (10)

Now we first consider the case n =5. Consider Fact 2 for [alpha] = p, [beta] =(- sin [theta], cos [theta], 0), and [gamma] = [a.sub.3]. Then there exists x [member of] R such that

[a.sub.3] = (cos [theta]) p + (sin [theta] cos x) (- sin [theta], cos [theta], 0) + (sin [theta] sin x) (0, 0, 1). (11)

Next, we consider Fact 2 for [alpha] = [e.sub.1], [beta] = (0, 1, 0), and [gamma] = [a.sub.5]. Then there exists z [member of] R such that

[a.sub.5] = (cos [theta], sin [theta] cos z, sin [theta] sin z). (12)

Finally, we consider Fact 2 for [alpha] = [a.sub.5] in (12),

[beta] =(- sin [theta], cos [theta] cos z, cos [theta] sin z), (13)

and [gamma] = [a.sub.4]. Then there exists y e R such that

[a.sub.4] = (cos [theta])a + (sin [theta] cos y) [beta] + (sin [theta] sin y) (a x [beta]). (14)

Now we define the function f : [(R/2[pi]Z).sup.3] x [0, [pi]] [right arrow] R by

f(x, y, z, [theta]) :+ <(11), (14)> - cos [theta]. (15)

We can understand [M.sub.5]([theta]) as a level set. More precisely, we define the function

h : [f.sup.-1] (0) [right arrow] R (16)

by h(x, y, z, [theta]) = [theta]. Then we have

[M.sub.5]([theta]) = [h.sup.-1]([theta]) (17)

if 0 < [theta] [less than or equal to] [pi].

Remark 3. Since f(x, y, z, 0) = 0 for all x, y, and z, we have [h.sup.-1](0) = [(R/2nZ).sup.3]. On the other hand, it is clear that [M.sub.5](0) = {one point}. Hence (17) does not hold for [theta] = 0. Apart from this point, there is an identification

y \ [M.sub.5](0) = [f.sup.-1] (0) \ [h.sup.-1] (0), (18)

where Y is defined in (6).

Lemma 4. We set

[mathematical expression not reproducible]. (19)

Then S is given in Table 4.

Proof. The lemma is proved by direct computations.

Proof of Theorem C. We consider h in (16) as a Morse function on [f.sup.-1](0). First, Table 4 and (17) show that [M.sub.5](4[pi]/5) = {one point}.

Second, direct computation shows that

[mathematical expression not reproducible]. (20)

Since this is nonzero, the space [f.sup.-1](0) is smooth at (0,0, n,An/5). Actually, we can prove that the point is a nondegenerate critical point of the function h. Hence Morse lemma shows that there is a homeomorphism [M.sub.5](0)[congruent to] [S.sup.2] for 2[pi]/3 < [theta] < 4[pi]/5. But if we use [11, Corollary B], we need not check that h is nondegenerate at (0, 0, [pi], 4[pi]/5). For our reference, we draw the figure of [M.sub.5](4[pi]/5 - 0.1) in Figure 3.

Third, the other parts of Table 3 follow from Table 4. This completes the proof of Theorem C.

Proof of Corollary D. The corollary is an immediate consequence of Theorem C.

Proof of Theorem B. We define [a.sub.3] as in (11). We also define [a.sub.4] to be the right-hand side of (12). We define the function f: [(R/2nZ).sup.2] x [0, [pi]] [right arrow] R by

f (x, z, [theta]) := <[a.sub.3], [a.sub.4]> - cos [theta]. (21)

Similarly to (17), we have [M.sub.4]([theta]) = [h.sup.-1]([theta]). Since [h.sup.-1]([theta]) is one-dimensional, it is easy to draw its figure. Thus Theorem B follows.

Proof of Theorem A. We define the function f : (R/2nZ) x [0, [pi]] [right arrow] R by f(x, [theta]) = <[a.sub.3], [e.sub.1]> - cos [theta]. Since [M.sub.3]([theta]) = [h.sup.-1]([theta]), Theorem A follows.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.


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Yasuhiko Kamiyama [ID]

Department of Mathematics, University of the Ryukyus, Nishihara-cho, Okinawa 903-0213, Japan

Correspondence should be addressed to Yasuhiko Kamiyama;

Received 3 November 2017; Accepted 1 February 2018; Published 1 March 2018

Academic Editor: Marc Coppens

Caption: Figure 1: (a) [M.sub.4]([theta]) for 0 < [theta] < [pi]/2 or [pi]/2 < [theta] < [pi]. (b) [M.sub.4]([pi]/2).

Caption: Figure 2: The Dupin cyclide.

Caption: Figure 3: [M.sub.5](4[pi]/5-0.1).
Table 1: The topological type of [M.sub.3]([theta]).

[theta]                                            Topological type

2[pi]/3 < [theta] [less than or equal to] [pi]        [empty set]
2[pi]/3                                               {one point}
0 < [theta] < 2[pi]/3                                {two points}
0                                                     {one point}

Table 2: The topological type of [M.sub.4]([theta]).

[theta]                   Topological type

[pi]                         {one point}
[pi]/2 < [theta] <[pi]       Figure 1(a)
[pi]/2                       Figure 1(b)
0 < [theta] < [pi]/2         Figure 1(a)
0                            {one point}

Table 3: The topological type of [M.sub.5]([theta]).

[theta]                               Topological type

4[pi]/5 < [theta] [less than            [empty set]
  or equal to] [pi]
4[pi]/5                                 {one point}
2[pi]/3 < [theta] < 4[pi]/5              [S.sup.2]
2[pi]/3                        Contains five singular points
2[pi]/5 < [theta] < 2[pi]/3         [[summation].sub.5]
2[pi]/5                         Contains one singular point
0< [theta] < 2[pi]/5                [[summation].sub.4]
0                                       {one point}

Table 4: The set S.

[theta]                    (x, y, z)

4[pi]/5                   (0, 0, [pi])
2[pi]/3    ([pi], 0, 0), (0, [pi], 0), (0, 0, [pi]),
                 ([pi], 0, [pi]), (0, [pi], [pi])
2[pi]/5                   (0, 0, [pi])
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Title Annotation:Research Article
Author:Kamiyama, Yasuhiko
Publication:Chinese Journal of Mathematics
Date:Jan 1, 2018
Previous Article:Corrigendum to "Geometric Framework for Unified Field Theory Using Finsler Gauge Transformation".

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