CONSTRAINING PLANE CONFIGURATIONS IN CAD

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CONSTRAINING PLANE CONFIGURATIONS IN CAD: GEOMETRY OF DIRECTIONS AND LENGTHS * WALTER WHITELEY t

Abstract. For;) configurl1tkm of phane points, eonstrnined by lenlths, there is on estAbllshed combinatorial and geometric theory of local uniqueness or 'rir:idlty of frameworks'. An equivalent theory of 'parallel drawings' exlsts for a conft~uratlon of plane points constrained by directions of lines. This paper presents an Initial a:eometric theory of dlrectlon·length designs which Incorporates and extends both of these previous theories. We present a 'klnemotic' theory of first.order uniqueness and robust designs, Including the conn«:tions among global. locol and ftrst-order uniqueness. The related 'static' analysis of designs and results on the Euclidean geometry of the singular POSitioM of robust designs completes the paper. This study ls both a natural completion of the theories of pArallel drowlngs and fint.order rigid plane framework and an opening Into a family of unsolved problems about angles ns constraints in CAD and elementnry geometry. Key words. constrained designs in CAD. length constrnints. rll:id frnmeworks. direction can· straints, parallel drnwings, robust designs. reciprocal designs.

AMS subject classification,. primary SIN05. 52C2S; secondary 65011. 68U01

1. Introduction. In plane Computer Aided Design (CAD). the basic object is a constramed design composed of: (a) a plane configuration (a collection of geometric objects in the plane): points, segments (lines) and arcs (circles), etc.; (b) constraints on the configuration: ti) prl::$cribed combinatOlial inctdences: points on lines, lines tar, gent to circle:, points Oil circles); (ii) dimensions (algebraic expressions in the coordinates or numbers): lengths between points, distances between points and lines, angles between lines etc. For a constrained design with a configuration of geometric objects and a set of constraints (incidences and dimensions). we have several related questions: (c) 00 the incidences and constraints determine a unique plane configuration (up to congruence) or a 'rigid configuration' (locally unique up to congruence)? (d) If the incidences and co~straints permit continuous deformations, which additional constraints would give the appropriate uniqueness? (e) Are the constraints indel'endent? Can one numerical dimension be changed or one incidence broken ane an appropriate 'nearby' design produced? (f) If the constraints are dep,·ndent. what are the minimal dependent sets (circuits)? Figure 1.1 illustrates a typic,,1 CAD design from 1151. In practice, all of these constraints can be expressed by a system of algebraic equations in appropriate variables for the geometric objects. Given an actual design. with the specific 'coordinates' for the geometric objects. the system of constraints can be linearized as a numerical matrix for the Jacobian of the algebraic system evaluated at the initial point 120,151. At

th~ l.' contains p', and p + p' is congruent to p - p', then p' is a trivial sllUke of FG(p). This reformulation implies that every shaky framework FG(p) has a sequence of pairs of arbitrarily close realizations where FG(p+ tp') "snaps" to an constraint equivalent design FG(p - tp'), since we can take tp' to be arbitrarily small. This justifies the term shaky. 0

The averaging technique also justifies the term 'robust'. We first need a some terminology do describe how changes in the dimensions (lengths or directions) of a design change the possible configurations (locally). While the direction represents a single 'dimension', it is usually expressed as a plane unit vector u, with U and -u identified - a point on the projective line !PI. For a double graph FG = (V; D, E), the dimension map f FG : lR2 1V1 - !p\DI x lRlEI is defined by: fFG(P)

= (... ,Uhi,···, Cik, ... )

=

=

where for {h,i} E D, Uhi I~~:~:I' and for {j,k} E E, Cik Ipi - Pkj2. Our constraint matrix for shakes is just the Jacobian of this dimension map (with rows mUltiplied by appropriate constants). ROBUSTNESS THEOREM 5.6. A direction-length design FG(p) is robltsti/, and only;/, the fn1/ouring two properties hold: (i) for the mtitial configuration p, there is an open neighborhood Np in which each configurahon PG( q) is tight; (ii) for the orig",al dimensions of FG(p),

d

= fFG(p) = (... , U/,i,"

.,cik, ... ) E !pIIDI x lRlEl,

and a fixed neighborhood Np of p, there is an open neighborhood Md of d such that for a1/ x E Md, there is a configuration q E Np with fFG(q) = x. Proof. Assume that FG(p) is robust. This means that the design FG(p) is stiff - and therefore tight by Theorem 5.4. Moreover, this stiffness is expressed by one polynomial inequality. If FG(p) is stiff there is an open neighborhood N p such that all q E Np satisify this same illequality. Therefore all these FG(q) are also stiff and tight. We change the domain of f FG to work with equivalence classes, modulo translation, lR2JVI /((t, ... , t) I t E lR2} = lR2JVI-2. This modified function is written iFG. The existence of (unique) configurations in f;~(x) is derived from the Inverse function Theorem applied to i FG as follows. (See [1,51 for the analogous application to lengths.) The constraint matrix (with the columns for one vertex dropped to remove translations) becomes the Jacobian of the dimension map i FG. Since FG(p) is robust, IDI + lEI = 21V1- 2 and the rank of this Jacobian is 21V1 - 2. We conclude that the Jacobian is square and invertible. The guarantees that the modified function iFG has an inverse on some open neighborhood Md of d = fFG(p}, with an image in Np • This inverse produ!".,s the required q for each x E Md.

19

GEOMETRY OF PLANE DIRECTIONS AND LENGTHS

Conversely, assume FG(p) is not robust. It is either shaky or FG(p) is stiff but Assume FG(p) has a non-trivial shake p'. By the modified version of the Averaging Theorem (Remark 5.5), for all t E !R, FG(p + tp') - FG(p - tp'), but p + tp' is not a translate of p - tp'. This creates a sequence of configurations contradicting the uniqueness in property (i) for all neighbohoods N p • We conclude that condition (i) implies FG(p) is stiff. Assume FG(p) is stiff. We know that jpc is injective in a neighborhood of p (by the Averaging Theorem). Property (ii) guarantees jpc is also onto, locally. Since jpc is an algebraic function, with domain !R2JVI-2, we conclude that the image in IP\DI x !RIEl has the same local dimension 2iVl- 2. However, property (ii) guarantees that IDI + lEI is the dimension of the image (locally). Thus IDI + lEI = 2iVi - 2 as required to make the stiff design robust. 0

IDI+IEI > 21V1-2.

Although we have the Swapping Theorem for stiff designs, tight designs may not swap to tight designs. Example 6.5 in the next section illustrates this failure. 6. Geometry of Special Positions. Robust double-graphs are robust for 'generic configurations' of the vertices. Previous experience with plane infinitesimal rigidity and with plane parallel drawings illustrate that the singular configumtions, where the underlying constraint matrix has reduced rank from the value at generic configurations, have an interesting geometry. For each of these previous cases of length designs and of direction designs, the set of 'singular positions' was projectively invariant. For the direction-length designs, the rank of the constraint matrix is only invariant only under similarity. The following example demonstrates there is no wider invariance: an affine transformation may change a stiff design into It shaky design.

o (fWI

A

(.. III

B

FIG. 6.1. ~ .subset fA) 0/ " gmericolly robu.st double graph (e) i.t positions ddermined by the EtJclillean !1t:ometn) o{ the configuration (8).

~ndent in .speCIal

EXAMPLE 6.1. Consider a parallelogram design, with alternating directions and lengths (Figure 6.1 A). For convenience, we assume that the points are (0,0), (0,1), (a,b), (a,b - 1). The constraint matrix is now:

[ IL

-1 0 0 6

0 1 1 -b a 0 0 0 0

0 b-I 0 0

0 -6

1 0

~I]

0 0 0 b - 1 -a

xu = O.

This has rank 4, if b is .. general value. However, if b = 1 (FIgure 6.IB), then the matrix has It row dependence:

(a , -I, -6, 1) [

~

-1 0 0 6

0 0 a 0 0 0 0

0 0 0 0

0 -a 1 0

0 0 0 0

~I] =

-a

O.

20

W . WHITELEY

and therefore has rank 3. Since the two designs in Figure 6.IA andd 6.18 only differ by an affine transformation, we conclude the special positions are not affinely invariant.

Figure 6.1 C shows the direction-length construction of a design containing this subgraph, confirming that generic configurations of the original double graph have rank 4.0 THEOREM 6.2. If two configumtions p, q on IVI points are similar, then the designs FG(p) and FG(q) have isomorphic spaces of shakes. In particular (a) FG(p) is shaky 'iJ. and only iJ. FG( q) is shaky; (b) FG(p lis robust iJ. and only iJ. FG( q) is robust; (c) FG(p) is independent iJ. and only iJ. FG(q) is independent. (d) FG(p) is tight iJ. and only iJ. FG(q) is tight.

Proof. We must show that the rank, independence etc. of any constraint matrix for a design FG(p) is not changed by a generating similarity map, q = T(p): a translation, dilation towards the origin, or a rotation about the origin. (a) A translation will not change any entries in the constraint matrix for the design. (b) A dilation by a factor A i' 0 will multiply the entries in the constraint matrix by this scalar A. This does not change the rank, independence etc. of the any submatrix. (c) Consider a rotation TQ by angle . If we rotate the shakes Ui by the same angle then: (Pi - pj)' (Ui - Uj) : TQ(Pi - Pj)' TQ(Ui - Uj)

and

(Pi - pj) " . (Ui - Uj)

= (TQ(Pi)

- TQ(pj))' (TQ(Ui) - TQ(uj»

= TQ(Pi -

pj) .l . TQ(Ui - Uj)

= (TQ(p;} - TQ(pJ» .l . (TQ(u;}

- TQ(uj».

We conclude that this rotation by induces an isomorphism of the shakes. (d) It also clear that translation, dilation and rotation take a loose path p(t) of designs to a loose path of designs, since rotation, congruence and dilation take parallel lines to parallel lines, and translation, rotation and dilation take equal length segments to equal length segments, as required for a loose path. 0 The arguments in [7,221 showed that the special positions of an isostatic graph G for bar frameworks (the analog of a robust double-graph in §2) are characterized by a single polynomial equation on the configuration: co(p) = O. While the matrix itself was not square, removing any 'valid' set of three columns and taking the determinant produced the same poly""mial, multiplied by a simple 'tie-down' factor which related to the chosen columns. (If we chose an 'invalid' set of columns, the determinant was identically zero.) An identical property and an analogous proof holds for these direction-length designs. PURE CONDITION THEOREM 6.3. Given a double gmph FG = (V; D, E) which has some realization as a robust design FG( q), the set of special configurations p for which FG(p) is not robust is chamcterized by a single polynomial equation CFO(p) = O. Up to sign, this polynomial is obtained by replacing the points by indetenninants, removing any first col-urnn for a vertex and any second column for a vertex (possibly the same) and taking the detenninant. The pure conl/ition CFO(X) is a homogeneous polynomial of degree 21V1 - 2 in the coordinates of the -vertices, and the (non · /tomogcneous) degree in the 'var;'ables Jor 'uerteL i /.'; at most the 'valence of i.

GEOMETRY OF PLANE DIRECTIONS AND LENGTHS

21

REMARK 6.4. It is clear that swapping a double graph leaves the pure condition unchanged. It is also clear, by standard arguments of [221, thllt the following substructures FG' = (V'; D', E') will produce homogeneous factors of the pure condition: (i) a subgrllph with ID'I + IE'I = 2JV' I - 2; (ii) a subgraph with ID'I = 2JV'I- 3; (iii) a subgraph with IE'I = 2JV'I- 3. 0 EXAMPLE 6.5. Consider the four point design which is constructed in Figure 6.2A. This contains an initial triangle abc of lengths. From the work on rigidity, we know that this piece has the pure condition [abcl where the bracket operation [.. ·1 represents the signed area of the triangle, or equivalently the determinant of the affine coordiates of the three points. This subdesign is singular if, and only if, the three points are collinear. This bracket is homogeneous of degree 2 and will be a factor of the pure condition of the whoie design. The remaining information is contained in a factor of degree 4.

flc. 6.2. TIlt. getlericaltv robusl double graph 0/ fA). or equl1Halenlly tM COJUtnnntJ 0/ three lengths nnd hvo angleJ of ( 8), is loose if Qnd only if the four point" are amcyclic (e).

We have attached three directions onto this triangle. Equivalently, up to rotation, I':e have fixed the angles Ladb, Lbdc (Figure 6.2 B). A standard theorem of Euclidean geometry~ says that moving cl with a constant Ladb, moves d on a circle through ti, b. Similarly, fixing angle Lbdc moves d on a circle through b, d. Therefore, all other designs d with matching angles will have d on a circle through a, b, c. We conclude that a necessary condition for two non-equivalent designs is the four points a, b, c, d lie on a circle (are conCTJclic) (Figure 6.2 e). This condition is also sufficient, since all d on a circle through n, b, c, give equal angles. Turning the triangle abc continuously to keep the direction of ad fixed gives a loose path p(t). Now this condition 'the (our points a,b,c,d lie on a circle' is a homogeneous, irreducible polynominl equation eta, b, c, d) = 0 of degree 4:

e(a,b,I,d)

= det

r?

xl

~

Xc

+Y? +y2

+ Yc~

x~ +Y~

x.

Y. Yb Xc Yc Xd Yd

X~

[J

=

o.

Therefore. up to a constant, this condition must be the entire residual factor of the pure condition. We conclude that the pure condition for this design is [abcle(a, b, c, d) = O. Of conrse, the swapped design FG8(p) has the sl1me pme condition (Figure 6.20). However, while four concyclic points gives a loose design for FG(p) in Figure 6.2C, the swapp'!(] design FG' (p) of Figure 6.20 is shaky, but tight. 0

7. Dependencies and Spans. For frameworks (constraints by length alone) there is a substantial literature on the static rigidity (row rank of the rigidity matrix) and seir-stresses (row dependencies)

22

W . WHITELEY

of the rigidity matrix. There is a comparable theory of row dependencies and row rank for the constraint matrix of a distance-length design. Since row rank equals column rank for any matrix, what is the advantage of such an equivalent theory? (a) Sometimes the row dependencies are easier to see. Implicitly, several of the arguments in previous sections used this point of view as a way to track the row rank of the constraint matrix. (b) With the dependencies on a planar graph, there is a geometric technique called the 'reciprocal diagram' (see §8). This generalization of the reciprocal diagram for frameworks [14,8,9J, gives additional geometric insight into the singular configurations described in the previous sections. We begin with the basic vocabulary for the row dependencies of the constraint matrix fo a distance length design. DEFINITION 7.1. A dependence on a distance-length design FG(p) is an assignment of scalars (i) W.b to the edges {a, b} E E, W.b "" Wbc and (ii) ~.b to the edges {a, b} E D such that ~.b = ~b4 and for each vertex a:

{bl{·.b} eE}

{ppOsite patch (i, h; k, j) -(fl, i;j, ") . If (h, i) e D, we say tbat the patches (h, i;j, k), (i, h; k,j) are direction patches in D . If {h,i} e E, we say that the patches (h,i;j,k), (i,h;k,j) are length patches in E.. We can now shift our assignments of scalars for a dependence from the edges to tbe edge-patcbes. We call this structure of patcbes S (V,U;D,E) an associated spherical polyhedron for the 2-connected planar double grapb FC. The associated spberical polyhedron for a planar double graph has certain basic properties, wbich we state without proof (see IS,nil:

=

=

PROPOSITION 8.1. Gi','en" 2-connected planar double graph FC = (V;D,E), an associated spherical pOlyhedron S ~ (V, U; D, E) has the /o/l01l1ing properties:

26

w.

WHITELE\,

(t) If I, = (h, i; j, k) is In 12 (Tesp. E.), then the companion edge potch -b = (i,lt; k, j) is in D (resp. ill but (h,i;k,j) and (i,lt;j,k) aTe not in D (Tesp. E); (ii) FaT each veTtex v, all the edge patches in DUg with fiTst 'uertex v form a cycle: (v,i,;i',k') witlt 1 S sSt and t ~ 2; witltk' = i'+' andj' = k', and with all the edges patches are rlistin.ct;

Dug with fiTst face f fOTm a cycle: with is = h,,+1 and i( = h1, and with all

(iii) FOT each face f, all the edge potches in (h",i,,;/,k"), with 1:5 s 5 t and

t:5 2;

the edge patches are distinct; (iv) FOT each po·iT of veTtices n, b, there is a connecting vertex-edge path: a sequence of patches in 12 U E, (hr, iT; jr, kr) faT 0 S T S t, such that hr+ 1 = iT and ho = a,

i,

= bi

(v) FOT each pair of faces c, d, theTe is a connecting face-edge poth: a sequence of patches in D u £ (Its I itl;j", k") JOT 0 :5 s :5 tt such that j.J+l = k lJ I jO = c, k' d; (vi) Each simple 'uertex-edge cycle (closed path) separates the faces into two face-edge path-connected components; (vii) Each simple face-edge C1Jcle (closed path) separates the vertices into two vertexedge path-connected components.

=

REMAnK 8.2 . This associated spherical polyhedron is an abstract structure produced by some initial topological planar embedding. It does not depend on the particular configuration used for a distance-length design (which may not be planar or may be planar with different 'regions'.) ThrougllOut tbis paper we work with tbis topology (/lOmology) of tbe combinatorial polyhedron to define cycles etc. a

If we switch the vertices and faces in these patches: (h, i;i, k) ... 0, k ; h, i), we all .ll;sociated dual polyhedron S· - (U, V; D· , E·). All the properties listed in P"oposition 8.1, except (vii ) apPPar in pairs which now apply to the dual polyllL'dron. That property (vii) is self dual, in the presence of the other properties, is a th~'Orem of topology. Unfortunately, our assumptions do not guarantee that this dual polyhedron comes from It (dual) double graph. In the original polyhedron, two 'faces' may share many edges (of the same type). The dual polyhedron comes from a double multi-graph in which the sets D' and E' may have several edges joining two vertices. As long as we label these by the original patches, we have distinct labels and no confusion wiII result. The render can verify that everything in this paper works for double multi-graphs as well as double graphs. The main reason for avoiding these is that two edges of the same type, joining the same vertices, will produce identical constraint equations - and be inhm'ently dependent (redundant). For purposes of the geometric (combinatorial) construction of the reciprocals it is convenient to allow such objects - and we will. As necessOl'y, we will implicitly work with such 'multigraph' designs S(p) and S'(q) on the associated polyhedron and its dual. produc~

DEFINITION 8.3 . Given a direction-length design S(p) with associated polyhedron S = (V, U; D, E), a reripmcal direction-lengt" design is "direction-length design on the dual polyhedron S·(q) , such that: (i) for each edge patch (It, i;j, k) E D, qj - qk is perpendicular to Ph - Pi; (ii) for each edge patch (It, i;j, k) E g, qj - qk is parallel to Ph - PI .

Notice that (S·)· = S, .1Ild the definition makes S a reciprocal desib'll to S·. This definition produces a tr"e 'nciprocal pairing' of S(p) and S·(q) .

27

GEOMETRY OF PLANE DIRECTIONS AND LENGTHS 2

0

2

, J

A

B

A

r

2 "'_:--e. • 4

O

c

D

Flo. 8.3. A pair 0/ recipf"'OClll de$lgm (A,D) with the local coJTe.tpondencu for a length (8) and a dinction (e).

EXAMPLE 8.4. Consider the design in Figure 8.3 A. We have drawn it with the two directions (1,2) and (3,4) perpendicular. As we shall see, this makes the design dependent because there is a reciprocal. The design of Figure 8.30 is such a reciprocal, corresponding to a full dependence. The two designs look the same, but the labels are essential different, giving perpendicular dual edges for the directions, and parallel dual edges for the lengths. 0 THEOREM 8.5. For a direction-length design with an associated polyhedron S(p), the following are e 21V1-2, we are guaranteed there is a non-trivial dependence. Moreover, ID'I+IE'I::; 21V'1-2 for all proper subsets, (and ID'I::; 21V'1-3 for all nonempty subsets D' and similarly for subsets E') . Therefore, we arc guaral1teed that at II. generic configuration this double graph will have a full dependence. There must be a reciprocal design, unique up to choice of scale and initial vertex (Fig Jre 8.8 B). Notice that the dual has ID'I + IE'I < 21FI - 2. For generic configurations this dual double graph is independent but it is always shaky. Since the reciprocal is dependent, it has a very special geometry. Il As the previous example indicates, the counts IDI + lEI = 21V1- 2, ID'I + IE'I ::; 21V'1-2Iie at a critical point for the reciprocal construction. Since we have assumed a connected planar graph, Euler's formula gives IVI- (IDI + IE\) + IFI = 2. Multiplying by 2 and separating terms, we have (i) IDI + lEI = 21V1- 2 if, and only if, ID'I + IE'I = 21F1- 2; (ii) IDI + lEI> 21V1- 2 if, and only if, ID'I + IE'I < 21F1- 2. A more subtle check of the Euler characteristic yields, in case (i) that ID'I + IE'I < 21V'1 - 2 for proper subgraphs if, Ilnd only if, ID'''I + IE"'I < 2JF"1 - 2 for proper subgraphs of the dual. The original double graph is generically robust if, and only if, the dual double graph is generically robust. REMARK 8.11. Many double graphs are not planar. However, there are standard 'tricks' used for frameworks, related to an old engineering technique called 'Bow's notation' 181. Essentially, provided that two edges cross in a general way (in their interior, not parallel when they arc both directions or both lengths, and not perpendicular when one is a direction and the other a length) then we can insert a new vertex at the crossing point, splitting the two edges, to get a new design which has an isomorphic space of dependencies and "f shakes. This lets us turn many designs with non-planar graphs into eqUivalent desil:ns with planar graphs, which can then be analysed with reciprocals.

GEOMETRY OF PLANE DIRECTiONS AND LENGTHS

31

REMARK 8.12. We note that our reciprocals, with parallel edges for length constrnints, correspond to the classical construction of Cremona \10\, rather than the classical construction of Maxwell \14\. In our setting, Maxwell's reciprocal construction corresponds to a dependence on a purely direction design. For length constraints alone (frameworks) the reciprocal diagrams are intimately connected with projections of spherical polyhedra from 3-spaco, with distinct planes for faces meeting at an edge \8,9,14\. Specifically, the reciprocal pair corresponds to a projection of a spatial polyhedron and a specific spatial polar \8,9\. That connection was based on the dependencies satisfying the addional 'moment equation' around closed cycles, (related to the rows of the constraint matrix being orthogonal to the rotations of the configuration). For direction-length designs, the reciprocals have no interpretation in terms of objects in 3-space. We have reached a natural limit to the geometric theory connecting reciprocal pairs and spatial polyhedra. 0 Acknowledgements. This new theory brings out many insights and results derived from my long term collaborations on the rigidity of frameworks and parallel draWings with Janos Baraes, Henry Crapo, Bob Connelly, Tiong-Seng Tay, and Neil White. It is no longer possible to be sure when, or where, a particular connection within this geometry arose. Conversations with John Owen focused my attention on some related problems in CAD and began the train of thought which generated these results. Recent joint work with Brigitte Servntius clarified the combinatorial basis and the presentation of this work. Work with four summer undergraduate researchers: Melissa Bousfield, Katherine Caldwell, Linh Duong and David Moskovitz placed the connections with the original problem of plane angles and lengths into a clearer perspective \3\ and clarified my own thinking in a numbe,' of ways. Camille Cooper assisted both with the work of the undergraduate researchers and with the presentation of this paper.

REFERENCES

II) L.

ASIMOW AND

B.

ROTH,

The rigidity 0/ gmpll~, Trans. Amer.

Math. Soc. 245 (1978),

27D-289.

121 J.

Computational Synthetic Geometry, Spingcr-Vcrlag 1989. D. MOSCOVITZ AND W. WHITELEY, ConsLraintng Plane Conjigu1'Ohoru in CAL. Angles, to 3ppcar. 141 R. CONNELLY, Btuic conapts of ttgidity, draft chapter (1988) (or The Geometry of RI~ld Structures, H. Crapo and W. Whiteley (eds.), to appear. 151 - - , BasiC concepts of infinite.iimal rigidity, draft cilal)tt!f (1988) (or The Geometry of Rigid Structul"C:3, H. Crapo and W. Whiteley (eels.), to appear. 161 - - , Basic conapts of stotic r,! u1itV, draft chapter (1988) for Tho Ct.'Ometry of Rigid Structurcs, H. Crapo and \V. Whitl!lcy (t.·ds.). to nppear. 171 H. CRAPO. Tile cowbinntoriul th'nn) of .1tructures, in Mat.roid Tlk."Ory (Szcgcd 1982), NorthHolland New York, 1985, 101-213 181 H. CRAPO AND \;V. WIIITELEY, Plntl~ stT'l!.1se.1 and project~d polyhedra I: the bcuic pattern, Structural Topology 20 (1003). 55-78. 191 - - , S,-K1Cl!S of stresses. projections and parnUel drawtngs lor sphencal polyhedra, Contributkms to Algebra and Ceomt.... ry 35 (19901), 259~281. 1101 L. CREMONA, Graphical Statics, English translation, Oxford University Press, London, 1800, of

131 M.

BOKOWSKI AND B. STURMF&L5, BOUSFIELD, (0.:. CALDWELL,

L. DUONG,

the 1872 It..'\lian edition. GLUCI