相关论文: Surface Area of Ellipsoids

Consider a random smooth Gaussian field $G(x):F\to\mathbb{R}$, where $F$ is a compact in $\mathbb{R}^d$. We derive a formula for average area of a surface generated by the equation $G(x)=0$ and give some applications. As an auxiliary result…

We give a complete classification of Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo-sphere whose pseudo-spherical Gauss maps are of 1-type or, in particular, harmonic. In some cases a concrete global classification…

We give a characterization of all del Pezzo surfaces of degree 6 over an arbitrary field $F$. A surface is determined by a pair of separable algebras. These algebras are used to compute the Quillen $K$-theory of the surface. As a…

We deduce explicit formulae for the intrinsic volumes of an ellipsoid in $\mathbb R^d$, $d\ge 2$, in terms of elliptic integrals. Namely, for an ellipsoid ${\mathcal E}\subset \mathbb R^d$ with semiaxes $a_1,\ldots, a_d$ we show that…

Let $S$ be a complete flat surface, such as the Euclidean plane. We obtain direct characterizations of the connected components of the space of all curves on $S$ which start and end at given points in given directions, and whose curvatures…

Let $X$ be a quadratic vector field with a center whose generic orbits are algebraic curves of genus one. To each $X$ we associate an elliptic surface (a smooth complex compact surface which is a genus one fibration). We give the list of…

The expression for the variation of the area functional of the second fundamental form of a hypersurface in a Euclidean space involves the so-called "mean curvature of the second fundamental form". Several new characteristic properties of…

Various packing problems and simulations of hard and soft interacting particles, such as microscopic models of nematic liquid crystals, reduce to calculations of intersections and pair interactions between ellipsoids. When constrained to a…

In the calculation of thermodynamic properties and three dimensional structures of macromolecules, such as proteins, it is important to have a good algorithm for computing solvent accessible surface area of macromolecules. Here we propose a…

We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the Divergence Theorem to express the area and volume…

Consider a random set of points on the unit sphere in $\mathbb{R}^d$, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the…

In this paper, we study strong r-helix hypersurfaces and the special curves on these surfaces. Moreover, we investigated the relations between strong r-helix hypersurfaces and the Gauss transformations of these surfaces in Euclidean…

Extending classical results on polytopal approximation of convex bodies, we derive asymptotic formulas for the weighted approximation of smooth convex functions by piecewise affine convex functions as the number of their facets tends to…

Any permutation-invariant function of data points $\vec{r}_i$ can be written in the form $\rho(\sum_i\phi(\vec{r}_i))$ for suitable functions $\rho$ and $\phi$. This form - known in the machine-learning literature as Deep Sets - also…

In this paper, we study the special curves and ruled surfaces on helix hypersurface whose tangent planes make a constant angle with a fixed direction in Euclidean n-space Besides, we observe some special ruled surfaces in and give…

Let $f$ be a function that is analytic in the unit disc. We give new estimates, and new proofs of existing estimates, of the Euclidean length of the image under $f$ of a radial segment in the unit disc. Our methods are based on the…

We shall discuss cosmological models in extended theories of gravitation. We shall define a surface, called the model surface, in the space of observable parameters which characterises families of theories. We also show how this surface can…

We present a general algorithm for finding the overlap area between two ellipses. The algorithm is based on finding a segment area (the area between an ellipse and a secant line) given two points on the ellipse. The Gauss-Green formula is…

In this paper we define the magnitude of metric spaces using measures rather than finite subsets as had been done previously and show that this agrees with earlier work with Leinster in arXiv:0908.1582. An explicit formula for the magnitude…

The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the image of a non-closed geodesic has 0 distance from the set of conical points.…