Universality Theorems for Connguration Spaces of Planar Linkages

arrangement Projective realization lx ly ld ly1 lx1 v 00 v 10 v 01 v 11 l∞ v x v y Lx Ly Ld L∞ (1, 0) (0, 1) (1, 1) (0, 0) (∞ , 0) (0, ∞) Line at infinity Figure 15: The standard triangle T and its standard realization. De nition 12.2 The standard realization T of the standard triangle T is determined by: T (v00) = (0; 0); T (vx) = (1; 0); T (vy) = (0;1); T (v11) = (1; 1) Here (0; 0); (1; 0); (0;1); (1; 1) are points in the a ne plane A 2 P2 which have the homogeneous coordinates: (0 : 0 : 1); (1 : 0 : 0); (0 : 1 : 0); (1 : 1 : 1) respectively. We say that an abstract arrangement A is based if it comes equipped with an embedding i : T ! A. Let (A; i) be a based arrangement. We say that a projective realization of A is based if i = T . Let BR(A;P2(k)) be the subset of R(A;P2(k)) consisting of based realizations, k = R; C . Lemma 12.3 (See [KM2, Theorem 8.20].) BR(A;P2(C )) is the set of complex points of a projective scheme over Z which is a scheme-theoretic quotient of R(A) by the action of PGL3. De nition 12.4 A functional arrangement is a based arrangement (A; i) with two subsets of marked point-vertices = (P1; :::; Pm) (the input-vertices) and point-vertices = (Q1; :::; Qn) (the output-vertices) such that all the marked vertices are incident to 43 the line-vertex lx 2 i(T ) (which corresponds to the x-axis) and such that the following two axioms are satis ed: Let BR0(A ) BR(A) denote the open subset which consists of realizations such that (Pj) 2 A 2 for all j, we de ne BR0(A ) similarly. Then we require: (1) BR0(A ) BR0(A ). (2) The projection p : BR0(A )! A m given by p( ) = ( (P1); :::; (Pm)) is an isomorphism of schemes over Z. Each functional arrangement determines a morphism f : A m ! A n (which is de ned over Z) by the formula: f(x) = q p 1(x) where q( ) = ( (Q1); :::; (Qn)). Theorem 12.5 (See [KM2, Lemma 9.7].) Let f : A m ! A n be any polynomial mapping with integer coe cients. Then there is a functional arrangement A which determines f . Let S A m be a closed subscheme de ned over Z, S = f 1(0) for some morphism f : A m ! A n . Let A be a functional arrangement which determines f as in the above theorem. By gluing the output vertices of A to v00 we obtain an arrangement A0 containing distinguished vertices P1; :::; Pm. Again de ne BR0(A0) by requiring (Pi) to be nite. We get an induced morphism (easily seen to be an embedding) p : BR0(A0) ! A m . We then have Theorem 12.6 (Theorem 1.3 of [KM2]) Let S be a closed subscheme of A m (again over Z). Then there exists a based marked arrangement A such that the input mapping p : BR0(A)! A m induces an isomorphism of schemes BR0(A)! S. We will use the following version of the above theorem: Theorem 12.7 Let X be a compact real algebraic set de ned over Z. Then there exists a based arrangement A such that X is entire birationally isomorphic to a Zariski open and closed subset C in BR(A;P2(R)). Proof: Using Theorem 2.18 we may assume that X is projectively closed. We choose the projective scheme X Pm whose set of real points is X so that the corresponding a ne scheme Xa A m+1 corresponds to a real reduced ideal. Thus X is Zariski dense in X(C ). De ne the a ne scheme XA = X \ A m . Now apply Theorem 12.6 to construct a based marked arrangement A so that BR0(A) is isomorphic to XA (as a scheme), hence the sets of real points of these schemes are isomorphic as well. Thus X is (polynomially) isomorphic to BR0(A; P 2(R)) which is Zariski open. It remains to show that BR0(A; P 2(R)) is also Zariski closed. Recall that BR(A) embeds canonically in a product (P2)N (P1)m where the last m factors correspond to the input vertices. The morphism p : BR0(A)! A m is the restriction of the projection on the last m factors: (P2)N (P1)m ! (P1)m The subset BR0(A;P2(C )) is constructible, hence its closure with respect to the classical topology is the same as its closure BR0(A;P2(C )) with respect to the Zariski topology in (P2(C ))N (P1(C ))m . 44 Suppose that there is a real point z 2 BR0(A;P2) that does not belong to BR0(A;P2(R)). Then z is the limit of a sequence zj 2 BR0(A;P2(C )). However p(zj) 2 Cm are obtained by \forgetting" all but the last m coordinates of zj , hence p(zj) will converge to a real point x of X. It is clear that x = 2 Rm and hence does not belong to X. This contradicts the fact that X is projectively closed. Remark 12.8 In general BR(A;P2(C )) is di erent from BR0(A;P2(C )). As an example consider the linkage A corresponding (via the construction in [KM2]) to the system of equations: x+ y = 0; x+ y = 1; x = y in A 2 . The set solutions of this system of equations is empty (even in the projective compacti cation of A 2). Thus BR0(A;P2(C )) = ;, on the other hand: BR(A;P2(C )) is a single point. Now we construct metric graphs corresponding to based abstract arrangements. Suppose that A is a based arrangement. We start by identifying the point-vertex v00 with the line-vertex l1, the point-vertex vx with the line-vertex ly and the point-vertex vy with the line-vertex lx in the standard triangle T . We also introduce the new edges [v10v00]; [v01v00]; [v10vx]; [v01vy] (Here v10; v00; v11; v01; ::: are the point-vertices in the standard triangle T .) We will use the notation L for the resulting graph. We construct a length-function ` on the set of edges e L as follows: 1) We assign the length =4 to the new edges. 2) We assign the length =2 to the rest of the edges. 13 The relation between the two universality theorems The goal of this section is to establish a relation between the two universality theorems for realizability of real algebraic sets (Theorems B and 12.7). Consider an abstract based arrangement A. We choose v00; vx; vy; v01; v10 as distinguished vertices of the corresponding metric graph L. Let L denote the metric graph L with the distinguished set of vertices as above. LetX be either S2 or RP2 with the standard metric d (so that the standard projection S2 ! RP2 is a local isometry). De ne the con guration space C(L;X) of realizations of L in X to be the collection of mappings from the vertex-set V(L) of L to X such that d( (v); (w))2 = (`[vw])2 for all vertices v; w of L connected by an edge. Remark 13.1 Notice that if a; b 2 RP2 are within the distance =2 then there are two minimal geodesics connecting a to b. This is the reason to de ne C(L;X) as the set of maps from V(L) rather than from L itself. One can easily see that C(L;X) has natural structure of a real algebraic set. The subsets M(L;RP2 ) := f 2 C(L;RP2) : (v00) = (0; 0); (vx) = (1; 0); (v10) = (1; 0); (v01) = (0; 1)g M(L;S2) := f 2 C(L;S2) : (v00) = (0; 0; 1); (vy) = (0; 1; 0); 45 (vx) = (1; 0; 0); (v10) = (1; 0; 1); (v01) = (0; 1; 1)g form cross-sections to the actions of the groups of isometries PO(3;R); O(3;R) of X on C(L;X). We callM(L;X), themoduli spaces of realizations of L inX (whereX = S2;RP2). Remark 13.2 Now it is convenient to use the full group of isometries of S2 instead of the group of orientation-preserving isometries that we used for planar linkages. Lemma 13.3 The moduli space M(L;RP2) is (polynomially) isomorphic to the real algebraic set BR(A;RP2). Proof: The key to the proof is the fact that a point P 2 RP2 is incident to a line L 2 (RP2)_ i d(P;L_) = =2 Thus we construct a morphism : BR(A0;RP2)!M(L;RP2); : 7! so that for each point-vertex P 2 A we have (P ) = (P ) and for each line-vertex L 2 A we have (L) = (L)_. This morphism has algebraic inverse given by the same formula (since (L_)_ = L). LetM0(L;RP2) be the image of BR0(A0;RP2) under the isomorphism . Consider the standard 2-fold covering S2! RP2 . It induces a (locally trivial) analytical covering :M(L;S2)!M(L;RP2 ) The group of automorphisms of is (Z2)r, where r is the number of (point) vertices in [L P(T )] [ fv11g. The generators of this group are indexed by the vertices v 2 [L P(T )] [ fv11g: gv : (v) 7! (v); gv : (w) 7! (w); w 6= v Proposition 13.4 For each arrangement A as in Theorem 12.7, the covering is analytically trivial over M0(L;RP2). Proof: The proposition will follow from the following: For each point-vertex v in L there is a line in RP2 and for each line-vertex v 2 L there is a line 0 in (RP2 )_ so that: (v) = 2 for all 2 BR0(A;RP2) (if v is a point-vertex) and (v) = 2 0 for all 2 BR0(A;RP2) (if v is a line-vertex). To prove this property recall (see [KM2]) that A is obtained from \elementary" arrangements for the addition and multiplication via ber sums. Thus it is enough to verify the above property for the arrangements CA; CM for the addition and multiplication that are described in [KM2]. The veri cation is straightforward and is left to the reader. Now we identify the moduli space of spherical linkages M(L;S2) with a moduli space of Euclidean linkages in R3 as follows: Add an extra vertex v0 to the graph L and connect it to each vertex of L by edge of the unit length. Modify the other side-lengths as follows: `0(e) :=p2 2 cos(`(e)); e 2 E(L) 46 Let L0 be the resulting metric graph with the distinguished set of vertices [P(T ) fv11g][fv0g. De ne the con guration spaceC(L0;R3 ) := f : V(L0)! R3 : j (v) (w)j2 = `0[vw]2gAgain is is clear that M(L0;R3) := f 2 C(L0;R3 ) : (v0) = (0; 0; 0);and the same normalization on P(T ) fv11g as we used for M(L;S2)gis a real-algebraic set which is a cross-section for the action of Isom(R3) on C(L0;R3).Obviously we have an isomorphismM(L;S2) =M(L0;R3)of real-algebraic sets. We letM0(L0;R3) be the subset of M(L0;R3) corresponding toM0(L;RP2 ) under the isomorphismM(L;RP2) =M(L;S2) =M(L0;R3 )Thus, as a corollary of Theorem 12.7 we obtain the following:Theorem 13.5 Let S be a compact real algebraic set de ned over Z. Then there are abstractlinkages L;L0 so that:(1)M0(L;RP2) is entire rationally isomorphic to S.(2)M0(L0;R3 ) is an (analytically) trivial entire rational covering of S.BothM0(L;RP2), M0(L0;R3) are Zariski open and closed subsets in the moduli spacesM(L;RP2), M(L0;R3) respectively.14 A brief history of \Kempe's theorem"This story began with the invention of the steam engine by Newcomen in 1722. One problemthat appeared naturally was to transform a periodic linear motion (of the \input" vertex) toa circular motion (of the \output" vertex). The \parallelogram" invented byWatt in the late18-th century gave an approximate solution to this problem. The \input" motion was notexactly linear, however the input vertex traces a curve with a point of zero curvature, hencethe output approximates a straight line up to the 2-nd order. After discovery in the rst halfof the 19-th century of several \unsolvable" geometric problems (like squaring a circle, etc.),for a while it was a common opinion that the problem of transforming linear to circularmotion also has no exact solution. This opinion was shared for instance by Chebyshev whoafter thinking about this problem introduced Chebyshev polynomials, partial motivation forwhich was the optimal approximate solution of the problem.This was the situation until 1864 when French navy o cer Peaucellier published a letter[Pe1] where he claimed a positive solution, without giving any details12.There are several opinions on what happened next (this caused a serious controversybetween Russian and French-British mathematical schools in the late 19-th century). In1871 Lippman Lipkin13 published the rst detailed solution [L]. Two years later (in 1873)12It seems that in 1860-s Peaucellier explained his solution to some other people, cf. [Ma], so his letter[Pe1] was probably not a hoax. However [Ma] contains only the title so we cannot be sure if Mannheimreally knew construction of the inversor.13That time Lipkin was a graduate student of Chebyshev. Lipkin had died in 1875 at the age of 25 fromthe smallpox.47 Peaucellier published a paper [Pe2] which also contained a detailed solution (the Peaucellierinversor) identical to Lipkin's. Immediately after that several other ways to \draw a straightline" were discovered [Ha], [Ke2]. As far as applications are concerned it turned out thatall the mechanisms that transform linear motion to circular are too complicated to be usedinstead of Watt's parallelogram, invention of e cient lubricants had closed the problem.The only practical application of the inversor we are aware of was in air engines whichventilated the British parliament in 1870-1880-s (see [W, Page 182]).The rest of the story is mostly pure mathematics. In 1875 A. B. Kempe published [Ke1]where (in the present terminology) he outlined a proof of the following theorem analogousto Theorem 11.2:Theorem 14.1 Suppose that S R2 is an algebraic curve, O 2 S. Then there exists anabstract closed C -functional linkage L, a Zariski closed algebraic subset C M(L) (whichis a union of irreducible components) and a closed 14 neighborhood U of O in S so that therestriction of the input map p to C is onto U .Remark 14.2 However, if one follows Kempe's arguments, C is not open inM(L), U 6= S(even if S is compact) and the mapping p : C ! U is not a trivial covering.Versions of Kempe's proof were reproduced in a number of places (see for instance [B]),however (as far as we know) even the assertion was not made precise and details of theproof were not given. Recently several (written) attempts were made to improve Theorem14.1, i.e. to make the subset C open and U = S (see [HJW]) and the projection pjCinjective, however, as far as we can tell, they were unsuccessful. Finally there was a workof W. Thurston on this subject that we have discussed in the Introduction.References[ABR] C. Andradas, L. Brocker, J. 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