Mathematical Research Seminar - Archive

In its classical formulation Plateau problem asks to find the surface of minimal area spanning a given boundary. Understanding the proper notion of surface, area and of "spanning a  boundary" has lead since the pioneering works of Almegren-De Giorgi/Federe-Fleming/Reifenberg in the 50's to the development of beautiful and powerful tools in Geometric Measure Theory.

I will review the state of the art on the Plateau problem by presenting a series of old and new results concerning existence and regularity of solutions.

In this talk we present a geometric approach to interpolate given sequence of rigid body positions (i.e. center positions and orientations), which, in contrast to standard approaches, is free of choosing parameter values in advance and it enables the lowest possible degree of the motion. We introduce some interpolation problems and develop interpolation schemes for the construction of rigid body motions by rational splines of a low degree. A slightly different approach to motion construction, namely motion design with Euler-Rodrigues frames of Pythagorean-hodograph curves is also discussed. Two presented schemes for motions of degree three and five  are particularly useful in the applications, where the orientational component is not precisely specified and an algorithm must be formulated to determine a "natural" variation of orientation along the center trajectory. The theoretical results are substantiated with numerical examples.

We prove several generalizations of ramarkable Pascal's theorem which states that if we draw a hexagon inscribed in a conic section then the three pairs of opposite sides of the hexagon intersect at three points which lie on a straight line - the Pascal line of the hexagon. The complete figure formed by 60 Pascal's line has amazing properties ant it is known as Hexagrammum Mysticum. The similar results about an octagon inscribed in a conic section are known as Octagrammum Mysticum. We present an elementary combinatorial proof of those classical using Bézout’s theorem as the main tool whose application is guided by the empirical evidence and computer experiments with the program Cinderella.