Asymptotes of Plane Curves - Revisited

In this paper we present a review of the basic ideas and results concerning asymptotic lines of plane curves. We discuss their different definitions, namely that of a limiting position of tangent lines, of the tangent line at infinity, and finally the one that requires that the distance between points of a curve and asymptotic line tends to 0 as the point moves along an infinite branch of the curve. We also recall the method of determining asymptotes of algebraic curves from the leading coefficients in their equation and provide examples.

Form-Finding of Shells Containing Both Tension and Compression Using the Airy Stress Function

Pure-compression shells have been the central topic in the form-finding of shells. This paper studies tension-compression mixed type shells by utilizing a NURBS-based isogeometric form-finding approach that analyzes Airy stress functions to expand the possible plan geometry. A complete set of smooth version graphic statics tools is provided to support the analyses. The method is validated using examples with known solutions, and a further example demonstrates the possible forms of shells that the proposed method permits. Additionally, a guideline to configure a proper set of boundary conditions is presented through the lens of asymptotic lines of the stress functions.

Form-Finding of Shells Containing Both Tension and Compression Using the Airy Stress Function

Pure-compression shells have been the central topic in the form-finding of shells. This paper studies tension-compression mixed type shells by utilizing a NURBS-based isogeometric form-finding approach that analyzes Airy stress functions to expand the possible plan geometry. A complete set of smooth version graphic statics tools is provided to support the analyses. The method is validated using examples with known solutions, and a further example demonstrates the possible forms of shells that the proposed method permits. Additionally, a guideline to configure a proper set of boundary conditions is presented through the lens of asymptotic lines of the stress functions.

Asymptotic Lines on the Pseudo-Spherical Surfaces

В трехмерном расширенном гиперболическом пространстве рассмотрим "полную" псевдосферу - поверхность вращения прямой вокруг параллельной ей прямой. Поверхность, лежащая в собственной области гиперболического пространства, локально несёт на себе геометрию плоскости Лобачевского. Одна часть ее вкладывается в евклидово пространство в виде хорошо известной воронки Бельтрами - Миндинга, другая вкладывается в трехмерное пространство Минковского в виде одного из псевдоевклидовых аналогов псевдосферы. Асимптотические линии на псевдоевклидовой части поверхности мнимы. Эти мнимые асимптотические линии можно интерпретировать как вещественные асимптотические линии на поверхностях с индефинитной метрикой постоянной кривизны. Для построения интерпретации привлекаются еще два псевдоевклидовых аналога псевдосферы Бельтрами - Миндинга. Один из них глобально изометричен продолжению "полной" псевдосферы за абсолют гиперболического пространства. В работе изучаются свойства асимптотических линий на рассматриваемых поверхностях постоянной кривизны с метрикой де Ситтера в~трехмерном псевдоевклидовом пространстве (пространстве Минковского). Эти свойства во многом аналогичны свойствам асимптотических на псевдосфере Бельтрами - Миндинга. Площадь сетевого четырехугольника асимптотической сети на поверхностях постоянной отрицательной кривизны евклидовом пространстве может быть найдена по формуле Хаццидакиса. В работе рассматривается аналог этой формулы для сетевых четырехугольников асимптотической сети на поверхностях постоянной кривизны с индефинитной метрикой в трехмерном псевдоевклидовом пространстве. Эта формула может быть обобщена на произвольные сетевые многоугольники асимптотической сети. С использованием индефинитной метрики можно распространить действие формулы Хаццидакиса за ребро псевдосферы Бельтрами - Миндинга.

On Lorentzian surfaces in ℝ2,2

We study the second-order invariants of a Lorentzian surface in ℝ2,2, and the curvature hyperbolas associated with its second fundamental form. Besides the four natural invariants, new invariants appear in some degenerate situations. We then introduce the Gauss map of a Lorentzian surface and give an extrinsic proof of the vanishing of the total Gauss and normal curvatures of a compact Lorentzian surface. The Gauss map and the second-order invariants are then used to study the asymptotic directions of a Lorentzian surface and discuss their causal character. We also consider the relation of the asymptotic lines with the mean directionally curved lines. We finally introduce and describe the quasi-umbilic surfaces, and the surfaces whose four classical invariants vanish identically.

Complete structure of the Fučík spectrum of the p-Laplacian with integrable potentials on an interval

To characterize the complete structure of the Fučík spectrum of the [Formula: see text]-Laplacian on higher dimensional domains is a long-standing problem. In this paper, we study the [Formula: see text]-Laplacian with integrable potentials on an interval under the Dirichlet or the Neumann boundary conditions. Based on the strong continuity and continuous differentiability of solutions in potentials, we will give a comprehensive characterization of the corresponding Fučík spectra: each of them is composed of two trivial lines and a double-sequence of differentiable, strictly decreasing, hyperbolic-like curves; all asymptotic lines of these spectral curves are precisely described by using eigenvalues of the [Formula: see text]-Laplacian with potentials; and moreover, all these spectral curves have strong continuity in potentials, i.e. as potentials vary in the weak topology, these spectral curves are continuously dependent on potentials in a certain sense.