Here we explore the geometry of the osculating spaces to projective varieties of arbitrary dimension. In particular, we classify varieties having very degenerate higher order osculating spaces and we determine mild conditions for the existence of inflectionary points. AMS Subject Classification: 14N05.

Here we present a partial generalization to higher order osculating spaces of the classical Lemma of Terracini on ordinary tangent spaces. As an application, we investigate the secant varieties to the osculating varieties to the Veronese embeddings of the projective plane. AMS Subject Classification: 14N05.

The object of this paper is to investigate the properties ruled surface which direction vector has a constant slope with osculator plane base curve in Galiean $3-$space. We obtain some kind by calculating geometric invariants. Also, we give an application on example and their graphs are visualized using Mathematica program.

Parabolas and paraboloids are used to introduce curvature, both qualitatively and quantitatively. 1. Introduction. Most of the standard diierential geometry textbooks recognize the oscu-lating paraboloid as a useful interpretation of the second fundamental form of a surface. This interpretation leads to some qualitative conclusions about the shape of the surface in consideration 9, pp.87,91]. T...

The projected deformation of stationary contours and markings on object surfaces is analyzed in this paper. It is shown that given a marked point on a stationary contour, an active observer can move deterministically to the osculating plane for that point by observing and controlling the deformation of the projected contour. Reaching the osculating plane enables the observer to recover the obje...

In this paper, we give a generalization of the osculating curves to $n$-dimensional Euclidean space. Based on definition an curve in 3 and 4 dimensional spaces, new type has been defined such that is independent ( n − ) (n−3) th binormal vector n-dimensional space, which called ”a generalized ”. We find relationship between curvatures for any unit speed be congruent E En . particular, character...

We describe a large-scale computational experiment studying structure in the numbers of real solutions to osculating instances of Schubert problems. This investigation uncovered Schubert problems whose computed numbers of real solutions variously exhibit nontrivial upper bounds, lower bounds, gaps, and a congruence modulo four. We present a family of Schubert problems, one in each Grassmannian,...

We present a general theory to obtain good linear network codes utilizing the osculating nature of algebraic varieties. In particular, we obtain from the osculating spaces of Veronese varieties explicit families of equidimensional vector spaces, in which any pair of distinct vector spaces intersect in the same dimension. Linear network coding transmits information in terms of a basis of a vecto...

In this note we introduce the notion of Grassmann convexity analogous to the wellknown notion of convexity for curves in real projective spaces. We show that the curve in G2,4 osculating to a convex closed curve in P is Grassmann-convex. This proves that the tangent developable (i.e. the hypersurface formed by all tangents) of any convex curve in P has the ‘degree’ equal to 4. Here by ‘degree’ ...