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Definition If to each value of the independent variable tTR called further by scalar argument to put the only vector of r (t) in compliance, r (t) call a vector function of scalar argument. A vector of r (t) from the beginning in the fixed point of O call radius-vektorm.

Let's remove a formula for calculation of curvature of this line in any its point of M (x, y). Let's assume thus that the curve is set in the Cartesian system of coordinates by the equation of a type of y=f (x) and that function has a continuous second derivative.

The curve can also be presented as the line of crossing of two surfaces with F1 equations (x, y, z) = 0, F2 (x, y, z) = Having chosen one of coordinates for parameter, it is possible to try to express through it from this system of the equations other coordinates. If it manages to be made, it will be possible to write down

. If the flexible inextensible thread fitting the set convex line L! to develop, keeping constantly tense, each its point will describe an evolvent of L. Therefore the evolvent is called still development. This operation of expansion of a thread is equivalent to swing without sliding of a straight line on this line L!; Each point of such straight line describes an evolvent of the L line L!. From this it follows that this evolute of L! has infinite number of evolvents of L. At the same time any this line considered as an evolvent has only one evolute.

Let's note that near various points the curve is bent differently. To characterize degree of a bend of this line in close proximity to this point And, we will enter concept of curvature of this point.

But y! there is a tangent slope to a curve in the corresponding point therefore follows from the received ratio that a tangent to a curve and a tangent to its evolute in the corresponding point are mutually perpendicular, that is the normal to a curve is a tangent to an evolute.

In introduction concepts of vector function were considered, leaning on which strict definition of a curve and its special case – a flat curve was given. In this point we will give definition of length of an arch and we will find its differential.

To the same point of a curve there can correspond various values of parameter t. Such points of a curve call it multiple points. Points about the radius vectors of r (a) and r (b) respectively are called as initial and final points of a curve. If the final point of a curve coincides with its initial point, a curve call closed. The closed curve which does not have multiple points at t (a, b) call the simple closed contour.

If this curve is defined by the equation of y=f (x), the equation (it is possible to consider as the parametrical equations of an evolute with parameter x. Excluding parameter x from these equations, we will receive direct dependence between the current coordinates of an evolute of  and . If the curve is set parametrically x = by  (t), y = (t), an uravneiya (give parametrical an evolute uravneiya.

The theorem If on some site M1M2 the curve radius of curvature changes monotonously, the evolute arch length increment on this site of a curve is equal on an absolute value to the corresponding increment of radius of curvature of this curve.