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# Evolute

## involute, curve, string and circle

EVOLUTE Awn INVOLUTE. See CURVATURE and OSCULATING CIRCLE. The evo lute of any curve is the locus of the center of its osculating circle, and, relative to its evolute, the curve is called the involute. This is the simplest definition that can be given of an evolute and involute, which are relative terms. There is another, however, which may represent the relation of the curves more clearly to those who are not mathematicians. If on any curve a string be closely wrapped, and if the string be fastened at one of its ends, and free at the other; and then if we unwind the string from the curve, keeping it stantly stretched, the curve which would be traced out by a pencil fixed to the free end of the string, is called the involute of that from which the string is unwound, and relative to it, the latter is called the evolute. It is clear that the involute might otherwise be described by ing a string at one extremity of the lute, and wrapping it thereupon, keeping it always stretched. From either tion, it is clear that a normal to the lute at any point is a tangent to the evolute, and that the difference in length between any two radii of curvature to the involute is equal to the length of the arc of the lute intercepted between them. The nature of evolutes was first considered by

ghens, who showed that the evolute to a common cycloid is another equal cycloid, a property of that curve which lie employed in making a pendulum vibrate in a cycloid. To describe the involute of a circle, proceed as follows: Let a be the center of the cle, and b the extremity of the string to be unwound from its circumference. Divide the circle, or part of the circle, according to the length of curve required, into any number of equal parts, as e, d, e, etc. ; thro'Ugh these, from a draw radial lines; from the points where these touch the circle, draw, at right angles to the lines ac, ad, etc., other lines, as in the diagram. With the distance eb as radius, from the point c, describe an arc bi, cutting the line e1 in 1. From the point d, with dl, describe an arc 1 2, cut ting the line d2 in 2. From e, with e2, describe an arc 2 3, cutting the line e3 in 3. With radius f3, from f, describe an are 3 4, cutting f4 in the point 4. Proceed in this way, . describing arcs which pass through the points 5, 6, 7, 8, and 9. The involute will thus be formed.