By Yu. Aminov

Differential geometry is an actively constructing quarter of contemporary arithmetic. This quantity offers a classical method of the overall themes of the geometry of curves, together with the speculation of curves in n-dimensional Euclidean house. the writer investigates difficulties for specific sessions of curves and offers the operating strategy used to procure the stipulations for closed polygonal curves. The facts of the Bakel-Werner theorem in stipulations of boundedness for curves with periodic curvature and torsion is additionally provided. This quantity additionally highlights the contributions made by means of nice geometers. earlier and current, to differential geometry and the topology of curves.

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L c shows a curve such that its singularity M is a cusp of the first type (the corresponding branches of the curve are situated on different sides with respect to the straight line containing the tangent ray to this curve at M); in the second case, M is a cusp of the second type (the tangent ray does not decompose the branches); in the third case, M is an isolated point; and in the fourth case, M is a point where two branches are tangent (in particular, these branches can coincide). If we assume that the first, second and third derivatives of the function p(x, y) are continuous, we can write Taylor's formula: + where o(Ar2) is an infinitesimal with respect to Ar2 = (x - x ~ )0,~- y0)2.

We have where the vector a; has the following form: Since every component of r' is a continuous function defined on the segment [a, h],it is uniformly continuous. Hence for any positive E there exists a positive 6 such that I a; I < E whenever I t , - t,-l 1 5 S. 2) it follows that Air - c = a,A,t. Therefore It is easy to obtain the following estimates: and Thus the difference between / Air I and Ir1(ri)lAlt is less than &Ai(. We apply the obtained estimate in order to estimate the difference between the length of y, and the sum C:=,Ir1(ri)/Ait: 4%) - The sum C:=,(rl(ri)lAitis a Riemann sum for the integral so the difference between this sum and integral is sufficiently small for a suitable choice of the points ti decomposing the interval [a,h].

A point M = (xo, yo) of the curve y is called a singularity of y,if the following conditions are fulfilled at this point: There exist different types of singularities. Assume that M = (xo,yo) is a singularity of y and some second derivatives of the function p do not vanish at M. Let us introduce some notations: There are three cases with respect to D: (a) if D > 0, then M is called an isolated point of y (Figure 9. lb); (c) if D = 0, then M is either an isolatedpoint, or a cusp, which can be of two types, or an osculate point of y (Figure 9.