The coordinates of the center of curvature belonging to the point of the curve are
If the plane curve is given in the parametric form , the corresponding parametric equations of the evolute are
In the spexial case that the curve is given in the form these equations can be written
For examining the properties of the evolute we choose for parameter the arc length , measured from a certain point of the curve; then in (1) the quantities and thus and are functions of . We assume that all needed derivatives exist and are continuous.
Differentiating (1) with respect to , we obtain
and recalling that , and it yields
If in the point of , the derivatives and do not vanish simultaneously, and so the evolute has in the corresponding point a tangent line with the slope
Since the of this is the slope of the normal line of the given curve , we have the
Theorem 1. The normal line of the curve in a point , where , is the tangent line of the evolute, having as tangency point the corresponding center of curvature . Thus the evolute is the envelope of the normal lines of the curve.
We shall calculate the arc length of the evolute corresponding the arc of the curve which is passed through when the parameter grows from to ; we assume that and are then continuous and distinct from zero. According the arc length formula,
Using the equations (2) and the fact that the sign of does not change, we can write
where and are the corresponding of . We have proved the
Theorem 2. The of an arc of the evolute is equal to the difference of the of the given curve touching the arc of the evolute in its end points, provided that and are continuous and do not change their sign on the arc of the curve.
- 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset I. Toinen painos. WSOY, Helsinki (1950).
|Date of creation||2013-03-22 17:35:06|
|Last modified on||2013-03-22 17:35:06|
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