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Motions and Stresses of Projected Polyhedra

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Motions and Stresses of Projected Polyhedra Motions and Stresses of Projected Polyhedra by Walter Whiteley* R&urn& Topologie Structuraie #7, 1982 Abstract Structural Topology #7,1982 L’utilisation de mouvements infinitesimaux de structures a panneaux permet Using infinitesimal motions of panel structures, a new proof is given for Clerk d’apporter une nouvelle preuve au theoreme de Clerk Maxwell affirmant que la Maxwell’s theorem that the projection of an oriented polyhedron from 3-space projection d’un polyedre de I’espace a 3 dimensions donne un diagramme plane gives a plane diagram of lines and points which forms a stressed bar and joint de lignes et de points qui forme une charpente contrainte a barres et a joints. framework. The methods extend to prove a simple converse for frameworks Les methodes tendent a prouver une reciproque simple pour les charpentes a with planar graphs - and a general converse for other polyhedra under appropriate graphes planaires - et une reciproque g&&ale pour ,les autres polyedres soumis conditions on the stress. The method of proof also yields a correspondence bet- a des conditions appropriees de contraintes. La methode utilisee pour la preuve ween the form of the stress on a bar (tension/compression) and the form of the pro&it aussi une correspondance entre la forme de la contrainte sur une bar dihedral angle (concave/convex). (tension/compression) et la forme de I’angle diedrique (concave/convexe). Les resultats ont une applicatioin potentielle a la fois sur I’etude des charpentes The results have potential application both to the study of frameworks and to et sur I’analyse de la scene (la reconnaissance d’images de polyedres). scene analysis (the recognition of pictures of polyhedra). I. Introduction I. Introduction En 1864, Clerk Maxwell a present& une theorie sur les diagrammes reciproques et sur In 1864 Clerk Maxwell presented a theory of reciprocal diagrams and equilibria in les equilibres des charpentes plans a barres et a joints. Cette theorie combinait des plane bar and joint frameworks. This theory combined older graphical methods for methodes graphiques plus anciennes pour les polygones de forces avec une nouvelle polygons of forces with a new geometric construction which used projective duality construction geometrique qui utilisait la dualite projective et la projection orthogonale and orthogonal projection to create both the framework and the ((reciprocal diagram, pour creer a la fois la charpente et le ((diagramme reciproqueu de forces dans les barres of forces in the bars from a single spatial polyhedron (Maxwell 1864, 1870). The basic a partir d’un seul polyedre spatial (Maxwell 1864, 1870). On peut resumer ainsi le result can be summarized: The projection of a spherical polyhedron gives a plane resultat fondamental: La projection d’un polyPdre sph&ique donne un diagramme diagram of lines and points which will have a static stress when built as a bar and joint plane de lignes et de points qui aura une contrainte statique lorsque construit comme framework. The size of the force in each bar is directly proportional to the length of the une charpente A barres et a joints. L’intensitti de la force dans chaque barre est direc- HreciprocalB line in the projection of a special dual polyhedron. tement proportionnelle a la longueur de la ligne &ciproqueN dans la projection d’un 13 poly&dre double spkial. Lorsque Maxwell, et plus tard, d’autres presentateurs proclamerent que la reciproque While Maxwell and later expositors claimed the converse was true (if a framework with etait vraie (si une charpente a graphe planaire possede une contrainte statique, elle est a planar graph has a static stress then it is the projection of a spherical polyhedron), no alors la projection d’un polyedre spherique), aucune preuve complete ne fut fournie. complete proof was provided. Nevertheless, the insight was sound and the resulting Neanmoins, leur point de vue etait bien fonde et les methodes de statiques graphiques methods of graphical statics were used by several generations of engineers (Culmann qui en resulterent furent utilisees par plusieurs generations d’ingenieurs (Culmann 1866, Cremona 1890, Henneberg 1911) before its virtual disappearance in the last 50 1866, Cremona 1890, Henneberg 1911) avant sa disparition reelle au tours des 50 der- years. Recently, following a conjecture of Janos Baracs, and several years of combined nieres annees. Recemment a la suite d’une conjecture de Janos Baracs, et de plusieurs research and detective work, a complete proof of the original result and its converse annees de recherches combinees et d’un travail de detective, on a developpe une was developed, using the original techniques (Crap0 & Whiteley, to appear). preuve complete du resultat original et de sa reciproque, en utilisant les techniques originales (Crap0 & Whiteley, a paraitre). Dans cet article, nous presentons une approche alternative a ce theoreme, basee sur In this paper we present an alternate approach to this theorem based on the general les techniques g&&ales de geometric projective pour les contraintes dans les charpen- projective geometric techniques for stresses in frameworks and motions of panel struc- tes et les mouvements de structures a panneaux presentees par (Crap0 & Whiteley tures presented in (Crap0 & Whiteley 1982). (We summarize these techniques in sec- 1982). (Nous resumons ces techniques a la section 2). Notre preuve du theoreme de tion 2). Our proof of Maxwell’s theorem begins with the observation that the process Maxwell debute par I’observation que le processus d’elevation d’une image plane a of lifting a plane picture up to the polyhedral scene in space is an infinitesimal motion une scene polyedrique dans I’espace constitue un mouvement infinitesimal de la struc- of the panel structure, composed of panels on the faces and hinges along the edges of ture a panneaux, composee de panneaux sur les faces et de charnieres le long des the flat projection (section 3.1). We then exploit a special correspondence between a&es de la projection plate (section 3.1). Nous utilisons alors une correspondance motions of a panel structure and stresses of a framework both built around an oriented speciale entre les mouvements d’une structure a panneaux et les contraintes d’une polyhedron. The result is a new proof of Maxwell’s theorem and its converse. charpente, les deux etant construits autour d’un polyedre orient& Le resultat constitue une nouvelle preuve du theoreme de Maxwell et de sa reciproque. Cette nouvelle vision kinematique produit plusieurs extensions importantes du champ This new kinematic vision yields several important extensions. First, the correspon- de recherche. Premierement, la correspondance entre les projections et les contrain- dence between projections and stresses directly extends to projections of other orien- tes amene directement aux projections d’autres polyedres orient&, pourvu que la sim- ted polyhedra, provided a simple path condition is satisfied by the stress (section 3). ple condition du chemin soit satisfaite par la contrainte (section 3). Nous obtenons aussi une correspondance plus detaillee entre le caractere des a&es We also obtain a more detailed correspondence between the character of the edges in du polyedre (convexe ou concave, aux limites ou a I’interieur dans la projection) et le the polyhedron (convex or concave, boundry or interior in the projection) and the type type de force dans la barre correspondante (tension ou compression) (section 4). Cette of force in the corresponding bar (tension or compression) (section 4). This split bet- division entre les membres de tension et de compression comporte des applications ween tension and compression members has important applications in the study of importantes pour I’etude des charpentes a tensegrite rigide (Roth & Whiteley 1981). rigid tensegrity frameworks (Roth & Whiteley 1981). Comme ce travail, aussi bien que le travail original de Maxwell, fut motive par les While this work, like Maxwell’s original work, was motivated by the problems of bar problemes de charpentes a barres et a joints, des parties de cette correspondance sont and joint frameworks, parts of this correspondence have also arisen in several recent aussi apparues dans plusieurs articles recents sur I’analyse de scene. En debutant avec papers in scene analysis. Beginning with the problem of recognizing when a picture of le probleme de reconna7tre lorsqu’une illustration de lignes et de points represente une lines and points represents a polyhedral scene, Huffmann recreated Maxwell’s theory scene polyedrique, Huffmann recrea la theorie de Maxwell sur les diagrammes of reciprocal diagrams (Huffmann 1977 a, b). Working on the same problem, Sugihara reciproques (Huffmann 1977 a, b). Ayant travaille sur le meme probleme, Sugihara fit pointed out the similar mathematical structure of the problems of scene analysis and ressortir la structure mathematique similaire des problemes d’analyse de scene et des the problems of moving frameworks (Sugihara 1980), and developed this further to give problemes de charpentes mobiles (Sugihara 1980), et developpa cette theorie pour a combinatorial version of Maxwell’s correspondence in the special case of frameworks arriver a une version combinatoire de la correspondance de Maxwell dans le cas with E = 2V - 2 edges, and vertices in ({general position)). special des charpentes a at-&es E = 2V - 2 et a sommes en ((position g&&ale)). 74 Cette histoire de redecouvertes frequentes augmente I’importance et l’attrait de la This history of frequent rediscovery emphasizes the importance, and the attraction of correspondance fondamentale. Dans le contexte de I’analyse de scene, I’approche the basic correspondence. In the context of scene analysis, the approach developed developpee ici nous amene a deux autres extensions. Notre construction projective here leads to two other extensions. Our projective construction leads to a new form of conduit a une nouvelle forme de diagramme reciproque basee sur le trace d’une set- .
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