Computer application of the kinematics control system for a virtual stretch-tightening press RO-630-11

A computer application of the control system for the virtual display of a real stretch-tightening press RO-630-11, taking into account its kinematic features, has been developed. The focus is on the research carried out by the finite element method in modeling the processes of shaping a sheet blank by a tightening punch, with a surface oriented relative to the curvature lines and placed on the table of a virtual stretch-tightening press RO-630-11.

The Ribaucour families of discrete R-congruences

While a generic smooth Ribaucour sphere congruence admits exactly two envelopes, a discrete R-congruence gives rise to a 2-parameter family of discrete enveloping surfaces. The main purpose of this paper is to gain geometric insights into this ambiguity. In particular, discrete R-congruences that are enveloped by discrete channel surfaces and discrete Legendre maps with one family of spherical curvature lines are discussed.

A graphical method to interpret how incision thresholds influence topographic and scaling properties of landscapes

<p>Theoretical analysis of the governing equations of numerical models can reveal relationships between topographic properties, such as drainage area, slope, and curvature, in simulated landscapes. These relationships are testable predictions; they can diagnose whether real-world landscapes could potentially arise from similar mechanisms. For example, the stream-power incision model is consistent with drainage area and slope data that plot as straight lines on logarithmic axes.</p><p>Here we graph theoretical relationships between topographic curvature and the steepness index, which depends on drainage area and slope. These relationships plot as straight lines for steady-state landscapes that have evolved according to a model that combines stream-power incision, linear diffusion, and uplift. Further, they link topography (drainage area, slope, and curvature) to characteristic length scales of the landscape, which depend on the competition between the processes of incision, diffusion, and uplift.</p><p>Adding an incision threshold to the model changes the relationship between the steepness index and topographic curvature. We examine these changes graphically and we show that they shed light on how incision thresholds influence topographic and scaling properties of landscapes. Specifically, we present a graphical method that consists of plotting steepness index–curvature lines and of tracing their intersections with each other and with the coordinate axes. This simple method reveals both how topography and process competition are influenced by the incision threshold, and how these influences vary within a given landscape and across different landscapes.</p>

Sticking Paint Replacement Films on Ship Hull Surfaced based on Lines of Curvature

In this paper, we propose a new concept of using the paint replacement films covering a ship-hull surface instead of paint based on lines of curvature. Firstly, we correct the B-spline hull surface within the prescribed deviation from the original surface shape based on nonlinear optimization to smooth the flow of curvature lines. After shape optimization, we adjust the position and the number of curvature lines so that it becomes the surface developments suitable for applying to thin sheet films. Using our method, it is possible to stick the films on the hull surface with few wrinkles. Finally, we applied proposed techniques to a bow of a bulk carrier to demonstrate the effectiveness of our algorithms.

Examples of surfaces of constant mean curvature

A surface in E3 is called parallel to the surface M if it consists of the ends of constant length segments, laid on the normals to the surfaces M at points of this surface. The tangent planes at the corresponding points will be parallel. For surfaces in E3 the theorem of Bonnet holds: for any surface M that has constant positive Gaussian curvature, there exists a surface parallel to it with a constant mean curvature. Using Bonnet's theorem for a surfaces of revolution of constant positive Gaussian curvature, surfaces of constant mean curvature are constructed. It is proved that they are also surfaces of revolution. A family of plane curvature lines (meridians) is described by means of elliptic integrals. The surfaces of constant Gaussian curvature are also described by means of elliptic integrals. Using the mathematical software package, the surfaces under consideration are constructed.

Cartan ribbonization and a topological inspection

We develop the concept of Cartan ribbons together with a rolling-based method to ribbonize and approximate any given surface in space by intrinsically flat ribbons. The rolling requires that the geodesic curvature along the contact curve on the surface agrees with the geodesic curvature of the corresponding Cartan development curve. Essentially, this follows from the orientational alignment of the two co-moving Darboux frames during rolling. Using closed contact centre curves, we obtain closed approximating Cartan ribbons that contribute zero to the total curvature integral of the ribbonization. This paves the way for a particularly simple topological inspection—it is reduced to the question of how the ribbons organize their edges relative to each other. The Gauss–Bonnet theorem leads to this topological inspection of the vertices. Finally, we display two examples of ribbonizations of surfaces, namely of a torus using two ribbons and of an ellipsoid using closed curvature lines as centre curves for the ribbons.