Development of Algorithms for Geometric Optimization and Data Analysis in Architecute

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Over the last three years, we have developed several geometric algorithms for optimization and data analysis in architecture. The summary of the results obtained as follows.1) We considered the problem of triangulating a convex polygon on spheres using n Steiner points that minimizes the overall edge length ratio. The problem arises in an application to approximation of curved surfaces of dome structures by triangular meshes. We establish a relation of this problem to a certain extreme packing problem. Based on this relationship, we develop a heuristic producing 6-approximation for spheres (provided n is chosen sufficiently large). That is, the produced triangular mesh is uniform in this respect.2) We studied the problem of rounding a real-valued matrix into an integer-valued matrix to minimize a n L_p-discrepancy measure between them. To define the L-p-discrepancy measure, we introduce a family of regions (rigid submatrices) of the matrix, and consider a hypergraph defined by the family. We first investigate the rounding problem by using integer programming problems with convex piecewise-linear objective functions, and give some nontrivial upper bounds for the L_p-discrepancy. We propose several interesting family of regions for which an efficient algorithm can be developed. We show that our approach is suitable for developing a high-quality digital-halftoning software.3) We developed an optimization method for finding an optimal floor layout of rooms, passages and door ways in a possibly non-rectangular site, based on mathematical programming as well as genetic algorithm.4) We developed a new method that quantitatively clarifies the relationship between the impression perceived on an photo of an architectural internal space and the phsical features of its color image. The effectiveness of the method was verified through questionnaires and experiments.