G. Khimshiashvili

Critical configurations of planar robot arms

It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, P can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive a formula for the Morse index of a critical configuration.

Equilibria of point charges on convex curves

We study the equilibrium positions of three points on a convex curve under influence of the Coulomb potential. We identify these positions as orthotripods, three points on the curve having concurrent normals. This relates the equilibrium positions to the caustic (evolute) of the curve. The concurrent normals can only meet in the core of the caustic, which is contained in the interior of the caustic. Moreover, we give a geometric condition for three points in equilibrium with positive charges only. For the ellipse we show that the space of orthotripods is homeomorphic to a 2-dimensional bounded cylinder.

Point Charges and Polygonal Linkages

We investigate the critical points of Coulomb potential of point charges placed at the vertices of a planar polygonal linkage. It is shown that, for a collection of positive charges on a pentagonal linkage, there is a unique critical point in the set of convex configurations which is the point of absolute minimum. This enables us to prove that two controlling charges are sufficient to navigate between any two convex configurations of a pentagonal linkage.

Cross-ratios of quadrilateral linkages

We discuss the cross-ratio map of planar quadrilateral linkages, also in the case when one of the links is telescopic. Most of our results are valid for a planar quadrilateral linkage with generic lengths of the sides. In particular, we describe the image of cross-ratio map for quadrilateral linkage and planar robot 3-arm.

Nondegeneracy of certain constrained extrema

The isolation and nondegeneracy of constrained extrema arising in geometric problems and mathematical models of electrostatics are studied. In particular, it is proved that a convex concyclic configuration of polygonal linkages is a nondegenerate maximum of the oriented area. Geometric properties of equilibrium configurations of point charges with Coulomb interaction on convex curves are considered, and methods for constructing them are presented. It is shown that any configuration of an odd number of points on a circle is an equilibrium point for the Coulomb potential of nonzero point charges. The stability of the equilibrium configurations under consideration is discussed.

Equilibria of three constrained point charges

We study the critical points of Coulomb energy considered as a function on configuration spaces associated with certain geometric constraints. Two settings of such kind are discussed in some detail. The first setting arises by considering polygons of fixed perimeter with freely sliding positively charged vertices. The second one is concerned with triples of positive charges constrained to three concentric circles. In each of these cases the Coulomb energy is generically a Morse function. We describe the minima and other stationary points of Coulomb energy and show that, for three charges, a pitchfork bifurcation takes place accompanied by an effect of the Euler’s Buckling Beam type.

The geometry of Kato Grassmannians

We discuss Fredholm pairs of subspaces and associated Grassmannians in a Hilbert space. Relations between several existing definitions of Fredholm pairs are established as well as some basic geometric properties of the Kato Grassmannian. It is also shown that the so-called restricted Grassmannian can be endowed with a natural Fredholm structure making it into a Fredholm Hilbert manifold.