By Yu. Aminov
Differential geometry is an actively constructing sector of contemporary arithmetic. This quantity offers a classical method of the final issues of the geometry of curves, together with the speculation of curves in n-dimensional Euclidean area. the writer investigates difficulties for exact periods of curves and offers the operating procedure used to acquire the stipulations for closed polygonal curves. The evidence of the Bakel-Werner theorem in stipulations of boundedness for curves with periodic curvature and torsion is usually provided. This quantity additionally highlights the contributions made via nice geometers. prior and current, to differential geometry and the topology of curves.
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Extra resources for Differential Geometry and Topology of Curves
If r(so)is the position vector of a point P, then the position vector p(so) of the curvature center 0 of y at P is the sum of the vector r(so) and the vector PO. Thus we obtain the position vector of the evolute: From the formula evolute is 7: = kv it follows that v', = -kr. 2). Assume that y is the evolute of some planar curve 7; then 7is called the evolvent of y. Let us find the position vector T; = r(s) of the evolvent with respect to the arc length s of y. 2 The curve y is the evolute of 7,hence its tangent vector r is collinear to the principal normal vector to the evolvent.
Suppose that there exists a sequence of points M, E y converging to M such that for some arbitrary fixed positive number k. 8) it follows that C = 0, at the points M,, o ( A r 3 ) / A y 2-+0 as n contradicting the assumption that C # 0. e. the curve y is tangent to a straight line collinear to the x-axis. Let us rewrite the equation of y: © 2000 CRC Press SINGULAR POINTS OF PLANE CURVES Then from this equation we obtain: If k l # 0, then the expression under the square root is equivalent to -4CklAx3 as Ar 4 0.
Now we will consider families of curves such that different curves of a family can be mutually intersecting. The equation of any such circle is For different a we have equations of different circles of the family. Two arbitrary sufficiently near circles are intersecting. In general we will assume that an equation defines a family of plane curves. 2) represents some curve y,,of the family. 1 © 2000 CRC Press 52 DIFFERENTIAL GEOMETRY A N D TOPOLOGY OF CURVES On examination of a family of plane curves, it can be seen that some plane curves are singular with respect to the family.
Differential Geometry and Topology of Curves by Yu. Aminov