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A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry is implicitly defined by the level sets of (one or more) multivariate polynomials. The algorithm recasts the implicitly…
In this paper, we study the rectifying curves in multiplicative Euclidean space of dimension 3, i.e., those curves for which the position vector always lies in its rectifying plane. Since the definition of rectifying curve is affine and not…
We consider skew ruled surfaces in the three-dimensional Euclidean space and some geometrically distinguished families of curves on them whose normal curvature has a concrete form. The aim of this paper is to find and classify all ruled…
With the rapid development of high-resolution 3D vision applications, the traditional way of manipulating surface detail requires considerable memory and computing time. To address these problems, we introduce an efficient surface detail…
We analyse a new notion of total anisotropic higher-order variation which, differently from the Total Generalized Variation by Bredies et al., quantifies for possibly non-symmetric tensor fields their variations at arbitrary order weighted…
A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete…
In this paper, we study unirational differential curves and the corresponding differential rational parametrizations. We first investigate basic properties of proper differential rational parametrizations for unirational differential…
This paper begins by extending the notion of a combinatorial configuration of points and lines to a combinatorial configuration of points and planes that we refer to as configurations of order $2$. We then proceed to investigate a further…
The usual approach to developing and analyzing first-order methods for smooth convex optimization assumes that the gradient of the objective function is uniformly smooth with some Lipschitz constant $L$. However, in many settings the…
Vector algebra is a powerful and needful tool for Physics but unfortunately, due to lack of mathematical skills, it becomes misleading for first undergraduate courses of science and engineering studies. Standard vector identities are…
Stationary and instationary Stokes and Navier-Stokes flows are considered on two-dimensional manifolds, i.e., on curved surfaces in three dimensions. The higher-order surface FEM is used for the approximation of the geometry, velocities,…
We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…
Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial…
Principal curves are natural generalizations of principal lines arising as first principal components in the Principal Component Analysis. They can be characterized from a stochastic point of view as so-called self-consistent curves based…
A subroutine for very-high-precision numerical solution of a class of ordinary differential equations is provided. For given evaluation point and equation parameters the memory requirement scales linearly with precision $P$, and the number…
The use of proximal point operators for optimization can be computationally expensive when the dimensionality of a function (i.e., the number of variables) is high. In this study, we sought to reduce the cost of calculating proximal point…
The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that…
Many parametrization and mapping-related problems in geometry processing can be viewed as metric optimization problems, i.e., computing a metric minimizing a functional and satisfying a set of constraints, such as flatness. Penner…
The Cartesian coordinate system is the most commonly used system in computer visualization. This is due to its ease of use and processing speed. However, it is not always suitable for a given problem. Angular measures often allow us to…
We derive pointwise curvature estimates for graphical mean curvature flows in higher codimensions. To the best of our knowledge, this is the first such estimates without assuming smallness of first derivatives of the defining map. An…