Construction of Hyperfigure Theory and Its Applications

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We found that the world of figures can be extended in such a way that the Minkowski operation always admits its inverse, and name the new objects "hyperfigures". The goal of this project was to study the mathematical structures of hyperfigures and establish a basis of hyperfigure theory. Aiming at this goal, we obtained the following results.(1) The object world is extended from the world of slope-monotone curves so that any polygons are included. This was achieved by the property that a slope-monotone curve can be the boundary of the union of a finite number of convex figures and that the polygonal lines are obtained as the limits of those curves.(2) Physical interpretation was given to hyperfigures. In the conventional Minkowski algebra, sometimes the result of the operation becomes just an empty set. In our new theory, such a result can be represent by a nontrivial hyperfigure, and can be interpreted as, for example, the ability of a numerical control machine or the ability of a water-spread vehicle.(3) An efficient algorithm was obtained for the Minkowski sum of three-dimensional solids. We considered surfaces of revolution obtained by rotating two-dimensional slope-monotone closed curves, and designed an algorithm for finding the correspondence of points with the same outward normal, and thus computing the Minkowski sum. This algorithm was also applied to the problem of finding the shortest distance of two moving solids.