| Abstract |
During the last 10-12 years there has been a dramatic revival of interest in applied geometric problems. Geometers have reconsidered a number of questions in infinitesimal mechanics, questions treated by J.C. Maxwell and L. Cremona [6, 12, 13] in 1864-70, further developed under the banner of graphical statics [7, 11], but left largely untouched since the end of the nineteenth century. At the same time, computer scientists have come to recognize that the tools of graphical statics and of applied projective geometry are fundamental to research in scene analysis.A good deal of the recent revival of interest is due to the efforts of the Structural Topology research group at the University of Montreal. The work of this group, reported in the pages of the journal Structural Topology [1, 2, 3, 4, 16, 17, 18] (and elsewhere), was a biproduct of research on infinitesimal mechanics, using methods derived from graphical statics, as well as from exterior algebra and its modern offspring, the Doubilet-Rota-Stein double algebra [8, 14]. Independently, Huffman [11], Duda and Hart [9], and others recognized that Maxwell's reciprocal figures could help in deciding whether a given plane image is the projection of a 3D polyhedral scene. More recently, Sugihara [15] and his colleagues in Nagoya created what may be considered a pilot project for automated descriptive geometry. They wrote a software package capable of modifying a rough plane sketch, so as to make it a true projection of a 3D scene.The starting point of the During the last 10-12 years there has been a dramatic revival of interest in applied geometric problems. Geometers have reconsidered a number of questions in infinitesimal mechanics, questions treated by J.C. Maxwell and L. Cremona [6, 12, 13] in 1864-70, further developed under the banner of graphical statics [7, 11], but left largely untouched since the end of the nineteenth century. At the same time, computer scientists have come to recognize that the tools of graphical statics and of applied projective geometry are fundamental to research in scene analysis.A good deal of the recent revival of interest is due to the efforts of the Structural Topology research group at the University of Montreal. The work of this group, reported in the pages of the journal Structural Topology [1, 2, 3, 4, 16, 17, 18] (and elsewhere), was a biproduct of research on infinitesimal mechanics, using methods derived from graphical statics, as well as from exterior a[8, 14]. Independently, Huffman [11], Duda and Hart [9], and others recognized that Maxwell's reciprocal figures could help in deciding whether a given plane image is the projection of a 3D polyhedral scene. More recently, Sugihara [15] and his colleagues in Nagoya created what may be considered a pilot project for automated descriptive geometry. They wrote a software package capable of modifying a rough plane sketch, so as to make it a true projection of a 3D scene. The starting point of the projective geometric analysis of scenes is the observation that the set of all three-dimensional realizations (scenes) having a given two-dimensional projection (a drawing, or image) form a linear space. Much information about an image, and about its possible spatial interpretations, can be obtained simply by calculating (either locally or globally) the linear dimension (or rank) of its linear space of scenes. In practice, the image is a pattern on a cathode-ray tube, an aerial photograph, an engineer's or architect's drawing, or an X-ray or NMR scan. The rank of its space of scenes will reveal whether there is ambiguity or uniqueness in the construction of its spatial interpretation, or whether such a construction is in fact impossible, as would be the case for a poorly conceived engineering drawing, or even in an otherwise correctly conceived drawing, if too many hypotheses are made concerning the 3D structure of the scene.Calculation of the rank of the space of scenes having a given image should, in principle, be accomplished using simple combinatorial algorithms based on easily-remembered rules-of-thumb. This is the goal, and it shows every sign of being achievable. The problem has, however, a certain degree of unavoidable difficulty. The requirement that a given image be an accurate projection of a non-trivial (non-planar) 3D scene imposes conditions on the image, conditions which are perhaps best described in terms of not-always-elementary constructions with straight-edge and pencil. In this paper, we begin to sort out the interplay of these projective conditions by creating a new homology theory for geometric configurations. The new homology theory applies to geometric objects which are more rigid, less pliable, than the “rubber sheets” studied by the topology of Henri Poincaré and his school. The passage to this higher degree of invariance is made possible by the creation of a homology theory with (restricted) vector, rather than (unrestricted) scalar, coefficients, or equivalently, by the use of a cohomology theory based on locally linear, rather than on locally constant, functions. We have verified that the new theory agrees with the cohomology theory for the sheaf of locally linear functions on a certain (combinatorially defined) topological space.The basic objects about which this new homology theory has something non-trivial to say are extremely general. From the geometric point of view, they are simply finite sets of points in a projective space or finite sets of vectors in a vector space. In order to emphasize the departure we take from linear algebra as it is usually practiced, we should say that we study vector spaces with a selected basis, that is, concrete vector spaces, in their usual representation as spaces FP of functions from a set P into a field F. Finally, we might say we are simply studying rectangular matrices. Since such objects are found throughout applied mathematics, the resulting homology theory has a very broad range of potential application. Indeed, potential applications of this new homology theory are to any domain where one is interested in the global behavior of systems determined locally by linear constraints. |
