Related papers: Generic distance-squared mappings on plane curves
A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. In this paper, we define naturally extended mappings of distance-squared functions, wherein each…
We define generalized distance-squared mappings, and we concentrate on the plane to plane case. We classify generalized distance-squared mappings of the plane into the plane in a recognizable way.
Any generalized distance-squared mapping of equidimensional case has singularities, and their singularity types are wrapped into mystery in higher dimensional cases. Any generalized distance-squared mapping of equidimensional case is not…
This is a survey article on distance-squared mappings and related topics.
In this paper, we discuss some problems of elementary plane differential geometry and kinematics. Although the results are not new, the consistent use of complex-valued functions (plane curves) of a real variable (parameter) allows to…
In the article, we exhibit a series of new examples of rigid plane curves, that is, curves, whose collection of singularities determines them almost uniquely up to a projective transformation of the plane.
A result is given to find points where a real valued function on the plane is not smooth. Provided this function is induced by a smooth mapping from three dimensions to the plane, from a function on surfaces in three dimensions. This has…
The field of multiple view geometry has seen tremendous progress in reconstruction and calibration due to methods for extracting reliable point features and key developments in projective geometry. Point features, however, are not available…
The generalized distance matrix of a graph is the matrix whose entries depend only on the pairwise distances between vertices, and the generalized distance spectrum is the set of eigenvalues of this matrix. This framework generalizes many…
We describe singularities of distance squared functions on singular surfaces in $\mathbb{R}^3$ parameterized by smooth map-germs $\mathcal{A}$-equivalent to one of $S_k$, $B_k$, $C_k$ and $F_4$ singularities in terms of extended geometric…
The order in which plane-filling curves visit points in the plane can be exploited to design efficient algorithms. Typically, the curves are useful because they preserve locality: points that are close to each other along the curve tend to…
We study deformations of plane curve singularities from an analytic point of view and obtain some new concrete results. We show some rather unexpected properties of Puiseux coefficients treated as functions on a suitably defined parameter…
We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the…
Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in…
Obtaining complete information about the shape of an object by looking at it from a single direction is impossible in general. In this paper, we theoretically study obtaining differential geometric information of an object from orthogonal…
We provide an elementary derivation of an orthogonal coordinate system for boundary layers around evolving smooth surfaces and curves based on the signed-distance function. We go beyond previous works on the signed-distance function and…
We investigate the geometry of holomorphic curves and complex surfaces from the perspective of singularity theory. We show that, with a suitable choice of a complex bilinear symmetric form, the families of functions and mappings that…
Algebras of generalized functions offer possibilities beyond the purely distributional approach in modelling singular quantities in non-smooth differential geometry. This article presents an introductory survey of recent developments in…
These notes are designed for those who either plan to work in differential geometry, or at least want to have a good reason not to do it. We discuss smooth curves and surfaces -- the main gate to differential geometry. We focus on the…
The Lorentzian length, which is one of the most significant functions in Lorentzian geometry, is a complex-valued function. Its square gives a real-valued non-degenerate quadratic function. In this paper, we define naturally extended…