Diameter, width and thickness in the hyperbolic plane

In hyperbolic geometry there are several concepts to measure the breadth or width of a convex set. In the first part of the paper we collect them and compare their properties. Than we introduce a new concept to measure the width and thickness of a convex body. Correspondingly, we define three classes of bodies, bodies of constant with, bodies of constant diameter and bodies having the constant shadow property, respectively. We prove that the property of constant diameter follows to the fulfilment of constant shadow property, and both of them are stronger as the property of constant width. In the last part of this paper, we introduce the thickness of a constant body and prove a variant of Blaschke’s theorem on the larger circle inscribed to a plane-convex body of given thickness and diameter.

Spore morphology of Taenitis, Syngramma, and Austrogramme species (Pteridoideae, Pteridaceae) from South-Eastern Asia and Oceania. II

This paper continues consideration of the spores of three paleotropical fern genera – Taenitis, Syngramma, and Austrogramme (Pteridoideae, Pteridaceae) from South-Eastern Asia and Oceania. At the second stage, we carried out a comparative scanning electron microscopy study of spores of three species of Austrogramme, four species of Syngramma, and six species of Taenitis and added information about previously studied spores of seven species of these genera. Spores of all examined species are trilete, tetrahedral or tetrahedral-globose with convex to hemispherical distal side and plane, convex or conical proximal side. The spores of Austrogramme species are the smallest, simplest in ornamentation and similar to each other. Sculpture of the proximal and distal sides are microverrucate, the surface of the spores is covered by granular deposits. Spores of most Syngrammaspecies are very similar to spores of Austrogramme species in shape and surface sculpture: their distal and proximal surfaces are microverrucate, whereas the spores of S. borneensis and S. cartilagidens have the low-tuberculate sculpture. Spores of Taenitis species are very different from the spores of Austrogramme and Syngramma. Seven of nine studied species have spores with well-expressed cingulum (T. blechnoides, T. cordata, T. diversifolia, T. interrupta, T. luzonica, T. obtusa, and T. requiniana), three species (T. cordata, T. hookeri, and T. pinnata) have spores with prominent laesural ridges. The spores have well-expressed ornamentation – tuberculate, baculate, rugate, tuberculate-rugate. The most conspicuous character of the ornamentation of spore surfaces is the presence of rodlets associated with sculpture elements. The densest rodlets are characteristic of Taenitis diversifolia, T. luzonica, T. obtusa, and T. requiniana. Spore size (equatorial diameter) ranges on average between 22 μm and 37 μm in Austrogramme, between 27 μm and 41 μm in Syngramma, and between 26 and 51 μm in Taenitis species.

Equilibria of Plane Convex Bodies

We obtain a formula for the number of horizontal equilibria of a planar convex body K with respect to a center of mass O in terms of the winding number of the evolute of $$\partial K$$ ∂ K with respect to O. The formula extends to the case where O lies on the evolute of $$\partial K$$ ∂ K and a suitably modified version holds true for non-horizontal equilibria.

Free Vibrations of Anisotropic Nano-Objects with Rounded or Sharp Corners

An extension of the Rayleigh–Ritz variational method to objects with superquadric and superellipsoid shapes and cylinders with cross-sections delimited by a superellipse is presented. It enables the quick calculation of the frequencies and displacements for shapes commonly observed in nano-objects. Original smooth shape variations between objects with plane, convex, and concave faces are presented. The validity of frequently used isotropic approximations for experimentally relevant vibrations is discussed. This extension is expected to facilitate the assignment of features observed with vibrational spectroscopies, in particular in the case of single-nanoparticle measurements.

Perimeter Approximation of Convex Discs in the Hyperbolic Plane and on the Sphere

Eggleston (Approximation to plane convex curves. I. Dowker-type theorems. Proc. Lond. Math. Soc. 7, 351–377 (1957)) proved that in the Euclidean plane the best approximating convex n-gon to a convex disc K is always inscribed in K if we measure the distance by perimeter deviation. We prove that the analogue of Eggleston’s statement holds in the hyperbolic plane, and we give an example showing that it fails on the sphere.