I. Kh. Sabitov

Surface deformation. I

This article surveys results on the theory of deformation and infinitesimal deformation of surfaces in three-dimensional Euclidean space that have been obtained in the last 25–30 years and do not appear in previous survey articles.

The rigidity of “corrugated” surfaces of revolution

The rigidity is proven of certain surfaces of revolution with infinite alternation of portions of positive and negative curvature.

On the Notion of the Combinatorial p-Parameter Property for Polyhedra

We introduce some versions of the notion of the combinatorial p-parameter property for polyhedra whose general meaning reduces to search for the number of parameters ensuring unique determination of a polyhedron locally on assuming given edge lengths and combinatorial structure.

On the relations between infinitesimal bendings of different orders

It is established that if a surface has only one nontrivial independent field of first order infinitesimal bendings, then rigidity of order n ≥ 3 implies rigidity of any order m ≥ n, in particular, nondeformability that is analytic in parameter. All the considerations are carried out in class C1.In passing, a new approach to the determination of infinitesimal bendings of higher order is given.

Approximation of plane curves by circular arcs

A method is proposed for approximating plane curves by circular arcs with length preservation. It is proved that, under certain rather mild constraints, any C3-smooth curve (open or closed, possibly, with self-intersections) can be approximated by a C1-smooth curve consisting of smoothly joined circular arcs. The approximation passes through interpolation nodes where it is tangent to the original curve, with the arc lengths between the nodes being preserved. The error of the approximation is estimated, and numerical examples are presented.

Configuration spaces of Bricard octahedra

We study the space of positions of a deformable octahedron and give a complete topological characterization of the structure of this space.

Possible generalizations of the Minagawa - Rado Lemma on the rigidity of a surface of revolution with a fixed parallel

A general necessary and sufficient criterion is established for the rigidity of a surface of revolution S ε C1 under the condition of a fixed parallel. Also two simple sufficient criteria for this property are given. It is shown by an example that this property does not hold for S ε C1 in the general case.

Locally Euclidean metrics with a given geodesic curvature of the boundary

The problem of reconstructing a locally Euclidean metric on a disk from the geodesic curvature of the boundary given in the sought metric is considered. This problem is an analog and a generalization of the classical problem of finding a closed plane curve from its curvature given as a function of the arc length. The solution of this problem in our approach can be interpreted as finding a plane domain with the standard Euclidean metric whose boundary has a given geodesic curvature.

On a class of inflexible polyhedra

We consider a class of polyhedra that we call pyramids and prove under some simple but rather general conditions on the extrinsic structure that the pyramids are inflexible. Moreover, this inflexibility property can be established also in multidimensional spaces of arbitrary constant curvature under appropriate conditions.

Isometric embeddings of locally Euclidean metrics in ℝ3 as conical surfaces

It is proved that if a domain with a locally Euclidean metric can be isometrically immersed in the Euclidean plane ℝ2 with the standard metric, then it can be isometrically embedded in ℝ3 as a conical surface whose projection on a sphere centered at the vertex of the cone is a self-avoiding planar graph with sufficiently smooth edges of specially selected lengths.