We compare different notions of curvature on contact sub-Riemannian manifolds. In particular we introduce canonical curvatures as the coefficients of the sub-Riemannian Jacobi equation. The main result is that all these coefficients are encoded in the asymptotic expansion of the horizontal derivatives of the sub-Riemannian distance. We explicitly compute their expressions in terms of the standa...

The liquid shape between two vertical parallel plates in a gravity field due to capillary forces is studied. When the physical system achieves its mechanical equilibrium, the capillary surface has mean curvature proportional to its height above a horizontal reference plane and it meets the vertical walls in a prescribed angle. We examine the shapes of these interfaces and their qualitative prop...

Metal curtain walls have been used to decorate the interiors and exteriors of large buildings. Most exteriors used in performing surface modeling of large buildings are linked or pieced by quadric surface. This surface, which is non-expansible, will cause wrinkling when stamped and used as a flat sheet metal of a metal curtain wall. These wrinklings are not distributed effectively because of th...

It is well known that no non-trivial Killing vector field exists on a compact Riemannian manifold of negative Ricci curvature; analogously, no non-trivial harmonic one-form exists on a compact manifold of positive Ricci curvature. One can consider the following, more general, problem. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorems canno...

It is shown that, in any secant polar stereographic projection, a small circle on a sphere projects into a circle. This property provides a simple relationship between K H , the horizontal component of curvature of a horizonal curve and Kb, the curvature of its projection on a secant polar stereographic map. K H can be computed by subtracting from the map factor times K h the earth’s curvature ...

Crash Modification Factors (CMFs) are used to represent the effects on crashes of changes to highway design elements and are usually obtained from observational studies based on reported crashes. The design element of interest for this paper is horizontal curvature on rural 2-lane highways. The data for this study came from the Washington State database in the Highway Safety Information System ...

A form-finding method is proposed for free-form shells specifying the distribution of stresses projected to horizontal plane. The surface shape obtained as a solution linear system vertical equilibrium equations discretized by finite difference with respect heights at grid points. Distribution shear stress specified generate various shapes region negative Gaussian curvature and free boundary. E...

Abstract We consider an asymmetric left-invariant norm ∥ ⋅ K {\|\cdot\|_{K}} in the first Heisenberg group ℍ 1 {\mathbb{H}^{1}} induced by a convex body ⊂ ℝ 2 {K\subset\mat...

Spatial relationship perception (SRP), defined as the function able to detect the difference between the perceived extent of a shape along the x/y cardinal coordinates, has been investigated in 42 eyes of 21 emmetropic subjects by means of a psychophysical test conceived on purpose. Aiming to the highest sensibility and since curvature detection is reckoned as an hyperacuity, elliptical stimuli...

We present the results of a combined theoretical and experimental investigation of the influence of surface tension σ on the laminar circular hydraulic jump. An expression is deduced for the magnitude of the radial curvature force per unit length along a circular jump, Fc =−σ (s − R)/Rj , where Rj is the jump radius, and s and R are, respectively, the arclength along the jump surface and radial...