Osculating surfaces of curves on surfaces

Oosculating sphere have been studied in classical differential geometry [1]. In this article the osculating surfaces of higher order of space curves on surfaces in Euclidean space is considered. We study the intrinsic differential geometry of curves on surfaces by analyzing their contact with surfaces of higher order.

The Use of Authentic Scientific Texts in the Process of Teaching Students to Solve Tasks of Differential Geometry

In the article presents the classification of problems by differential geometry, which is based on the nature of the relationship between the elements of the problem and the relationship between the reproducing and creative activity of students in their decision. It is shown that an important source for the choice of texts of problems and methods of their solution are the works of scientists – creators of classical differential geometry. Work with the corresponding scientific text allows the student to master such an educational strategy as methodological reduction.

Performance analysis of differential geometric guidance law against high-speed target with arbitrarily maneuvering acceleration

The interception of high-speed target with an arbitrary maneuvering acceleration causes serious troubles to the guidance and control system design of airborne missile. A novel guidance law based on the classical differential geometry curve theory was proposed not long ago. Although it is believed and numerically demonstrated that this differential geometric guidance law (DGGL) is superior to the classical pure proportional navigation (PPN) in intercepting high-speed targets, its performance has not been thoroughly analyzed. In this paper, using the Lyapunov-like approach, the performance of DGGL against the high-speed target with an arbitrary but upper-bounded maneuvering acceleration is well studied. The upper bounds of the LOS rate and commanded acceleration of DGGL are obtained, and conditions that guarantee the capture of this type of maneuvering target are also presented. The nonlinear relative dynamics between the missile and target is taken into full account. Finally, the proposed theoretical findings are demonstrated by numerical simulation examples.

A geometric method for eigenvalue problems with low-rank perturbations

We consider the problem of finding the spectrum of an operator taking the form of a low-rank (rank one or two) non-normal perturbation of a well-understood operator, motivated by a number of problems of applied interest which take this form. We use the fact that the system is a low-rank perturbation of a solved problem, together with a simple idea of classical differential geometry (the envelope of a family of curves) to completely analyse the spectrum. We use these techniques to analyse three problems of this form: a model of the oculomotor integrator due to Anastasio & Gad (2007 J. Comput. Neurosci. 22 , 239–254. ( doi:10.1007/s10827-006-0010-x )), a continuum integrator model, and a non-local model of phase separation due to Rubinstein & Sternberg (1992 IMA J. Appl. Math. 48 , 249–264. ( doi:10.1093/imamat/48.3.249 )).

A cubic surface of revolution

A well-known exercise in classical differential geometry [1, 2, 3] is to show that the set of all points (x, y, z) ∈ ℝ3 which satisfy the cubic equationis a surface of revolution.The standard proof ([2], [3, p. 11]), which, in principle, goes back to Lagrange [4] and Monge [5], is to verify that (1) satisfies the partial differential equation (here written as a determinant)which characterises any surface of revolution F (x, y, z) = 0 whose axis of revolution has direction numbers (l, m, n) and goes through the point (a, b, c). This PDE, for its part, expresses the geometric property that the normal line through any point of must intersect the axis of revolution (this is rather subtle; see [6]). All of this, though perfectly correct, seems complicated and rather sophisticated just to show that one can obtain by rotating a suitable curve around a certain fixed line. Moreover, to carry out this proof one needs to know a priori just what this axis is, something not immediately clear from the statement of the problem. Nor does the solution give much of a clue as to which curve one rotates.A search of the literature failed to turn up a treatment of the problem which differs significantly from that sketched above (although see [1]).The polynomial (1) is quite famous and has been the object of numerous algebraical and number theoretical investigations. See the delightful and informative paper [7].

Osculating curves and surfaces

Osculating circle and osculating sphere have been studied in classical differential geometry [1]. In this article the osculating curves and surfaces of higher order of plane and space curves in Euclideann-space (n = 2, 3) is considered. We study the intrinsic differential geometry of curves by analyzing their contact with curves and surfaces of higher order.

Osculating hypersurfaces of higher order

Oscurating surfaces of second order have been studied in classical differential geometry [1]. In this article we generalize this notion to osculating hyper-surfaces of higher order of hyper-surfaces inEuclidean n-space. Various related results are obtained using the derivatives of higher order.