69 research outputs found
Aukštesniųjų eilių glaustiniai paraboloidai
Osculating paraboloid of second order have been studied in classical differential geometry. In this article we generalize this concept to osculating paraboloids of higher order. This yields a visualization of the local properties of a given surface which depend on the derivatives of higher order.Šiame darbe nagrinėjami paviršiaus aukštesniųjų eilių glaustiniai paraboloidai afininėje koordinačių sistemoje. Tokių glaustinių paraboloidų panaudojimas leidžia analizuoti ir vizualizuoti duotojo paviršiaus lokalines savybes, kurios priklauso nuo aukštesniųjų eilių dalinių išvestinių
Singular Behavior of Electric Field of High Contrast Concentrated Composites
A heterogeneous medium of constituents with vastly different mechanical
properties, whose inhomogeneities are in close proximity to each other, is
considered. The gradient of the solution to the corresponding problem exhibits
singular behavior (blow up) with respect to the distance between
inhomogeneities. This paper introduces a concise procedure for capturing the
leading term of gradient's asymptotics precisely. This procedure is based on a
thorough study of the system's energy. The developed methodology allows for
straightforward generalization to heterogeneous media with a nonlinear
constitutive description
Envelope of mid-planes of a surface and some classical notions of affine differential geometry
For a pair of points in a smooth locally convex surface in 3-space, its
mid-plane is the plane containing its mid-point and the intersection line of
the corresponding pair of tangent planes. In this paper we show that the limit
of mid-planes when one point tends to the other along a direction is the
Transon plane of the direction. Moreover, the limit of the envelope of
mid-planes is non-empty for at most six directions, and, in this case, it
coincides with the center of the Moutard's quadric. These results establish an
unexpected connection between these classical notions of affine differential
geometry and the apparently unrelated concept of envelope of mid-planes. We
call the limit of envelope of mid-planes the affine mid-planes evolute and
prove that, under some generic conditions, it is a regular surface in 3-space.Comment: 15 pages, 1 figur
Characterizing envelopes of moving rotational cones and applications in CNC machining
Motivated by applications in CNC machining, we provide a characterization of surfaces which are enveloped by a one-parametric family of congruent rotational cones. As limit cases, we also address ruled surfaces and their offsets. The characterizations are higher order nonlinear PDEs generalizing the ones by Gauss and Monge for developable surfaces and ruled surfaces, respectively. The derivation includes results on local approximations of a surface by cones of revolution, which are expressed by contact order in the space of planes. To this purpose, the isotropic model of Laguerre geometry is used as there rotational cones correspond to curves (isotropic circles) and higher order contact is computed with respect to the image of the input surface in the isotropic model. Therefore, one studies curve-surface contact that is conceptually simpler than the surface-surface case. We show that, in a generic case, there exist at most six positions of a fixed rotational cone that have third order contact with the input surface. These results are themselves of interest in geometric computing, for example in cutter selection and positioning for flank CNC machining.RYC-2017-2264
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