Box-skeletons of discrete solids
Abstract The Medial Axis Transform (MAT) was defined by Blum in the 1960s as an alternate description of the shape of an object. Since then, its potential applicability in a wide range of engineering domains has been acknowledged. However, this potential has never quite been realized, except recently in two dimensions. One reason is the difficulty in defining algorithms for finding the MAT, especially in three dimensions. Another reason is the lack of incentive for modelling designs directly in MATs. Given this impasse, some lateral thinking appears to be in order. Perhaps the MAT per se is not the only skeleton which can be used. Are there other, more easily derived skeletons, which share those properties of the MAT which are of interest in engineering design? In this work, we identify a set of properties of the MAT which, we argue, are of primary interest. Briefly, these properties are dimensional reduction (in the sense of having no interior), homotopic equivalence, and invertibility. For the restricted class of discrete objects, we define an algorithm for identifying a point set, called a skeleton, which shares these properties with the MAT. Furthermore, this skeleton is to the box-norm ( L ∞ norm) what the MAT is to the Euclidean norm, and hence the deviation of this skeleton from the MAT is bounded. The algorithm will be developed for both 2D and 3D cases. Proofs of correctness of the algorithm shall be indicated. The use of this skeleton in automated numerical analysis of injection moulded parts shall be demonstrated on industrialsized parts. The use of the 3D skeleton in aiding automatic mesh generation for finite element analysis is also of interest, and shall be discussed.
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