EVOLUTE
, in the Higher Geometry, a curve first proposed by M. Huygens, and since much studied by the later mathematicians. It is any curve supposed to be evolved or opened, by having a thread wrapped close upon it, fastened at one end, and beginning to evolve or unwind the thread from the other end, keeping the part evolved, or wound off, tight stretched; then this end of the thread will describe another curve called the Involute. Or the same involute is described the contrary way, by wrapping the thread upon the Evolute, keeping it always stretched.
Thus, if EFGH be any curve, and AE either a part of the curve, or a right line; then if a thread be wound close upon the curve from A to H, where it is fixed, and then be unwound from A; the curve AEFGH, from which it is evolved, is called the Evolute; and the other curve ABCD described by the end of the thread, as it evolves or unwinds, is the Involute. Or, if the thread HD, fixed at H, be wound or wrapped upon the Evolute HGFEA, keeping it always tight, as at the several positions of it HD, GC, FB, EA, the extremity will describe the Involute curve DCBA.
From this description it appears, 1. That the parts of the thread at any positions, as EA, FB, GC, HD, &c, are radii of curvature, or osculatory radii, of the involute curve, at the points A, B, C, D.
2. The same parts of the thread are also equal to the corresponding lengths AE, AEF, AEFG, &c, of the Evolute; that is,
AE = AE | is the rad. of curvature to the point | A, |
BF = AF | " | B, |
CG = AG | " | C, |
DH = AH | " | D. |
3. Any radius of curvature BF, is perpendicular to the involute at the point B, and is a tangent to the Evolute curve at the point F.
4. The Evolute is the locus of the centre of curvature of the involute curve.
The finding the radii of Evolutes, is a matter of great importance in the higher speculations of geometry; and is even sometimes useful in practice; as is shewn by Huygens, the inventor of this theory, in applying it to the pendulum. Horol. Oscil. part 3. The doctrine of the Oscula of Evolutes is owing to M. Leibnitz, who sirst shewed the use of Evolutes in the measuring of curvatures.
To find the Evolute and Involute Curves, the one from the other.
For this purpose, put x = AD the absciss of the involute, y = DB its ordinate, z = AB the involute curve, r = BC its radius of curvature, v = EF the absciss of the Evolute, u = FC its ordinate, and a = AE a given line, (fig. 2 above). Then, by the nature of the radius of curvature, it is ; also by sim. triangles, , . Hence ; and ; which are the values of the absciss and ordinate of the Evolute curve EC; and therefore these may be found when the involute is given.
On the other hand, if v and u, or the Evolute be given: then, putting the given curve EC = s; since , or , this gives r the radius of curvature. Also, by similar triangles, there result these proportions, viz, , ; theref. , and ; which are the absciss and ordinate of the involute curve, and which may therefore be found when the Evolute is given. Where it may be noted that , and . Also either of the quantities x, y, may be supposed to flow equably, in which case the respective second fluxion x^{..} or y^{..} will be nothing, and the corresponding term in the denominator y^{.}x^{..}-x^{.}y^{..} will vanish, leaving only the other term in it; which will have the effect of rendering the whole operation simpler.
For Ex. Suppose it were required to find the Evolution EC when the given involute AB is the common parabola, whose equation is px=y^{2}, the parameter being p.
Here , making x^{..}=o. Then, to find first AE the radius of curvature of the parabola AB at the vertex, when x^{..}=o, the general value of the radius of curvature above given becomes =(by substituting the value of y^{.} and y^{..} &c) (―(p+4x)^{3/2})/(2√p) which is the general value of r or BC, the radius of curvature, for any value of x or AD; and when x or AD is = 0 or nothing, the value of r, or AE, becomes then only; that is half the parameter of the axis is the radius of curvature at the vertex of the parabola.
Again, in the general values of v and u above given, by substituting the values of y^{.}, y^{..}, and z^{.}, also o for x^{..}, and (1/2)p for a; those quantities become ; and . Hence then, comparing the values of v and u, there is found 3p^{1/2}v=4x^{1/2}u, and 27pv^{2}=16u^{3}; which is the equation between the absciss and ordinate of the Evolute curve EC, shewing it to be the semicubical parabola.
In like manner the Evolute to any other curve is found.—The Evolute to the common cycloid, is an equal cycloid; a property first demonstrated by Huygens, and which he used as a contrivance to make a pendulum vibrate in the curve of a cycloid. See his Horolog. Oscil. See also, on the subject of Evolute and Involute Curves, the Fluxions of Newton, Maclaurin, Simpson, De l'Hôpital, &c, Wolf. Elem. Math. tom. 1, &c, &c.
M. Varignon has applied the doctrine of the radius of the Evolute, to that of central forces; so that having the radius of the Evolute of any curve, there may be found the value of the central force of a body; which, moving in that curve, is found in the same point where that radius terminates; or reciprocally, having the central force given, the radius of the Evolute may be determined. Hist. de l'Acad. an. 1706.
The variation of curvature of the line described by the Evolution of a curve, is measured by the ratio of the radius of curvature of the Evolute, to the radius of curvature of the line described by the Evolution. See Maclaurin's Flux. art. 402, prop. 36.
Imperfect Evolute, a name given by M. Reaumur to a new kind of Evolute. The mathematicians had hitherto only considered the perpendiculars let sall from the Involute on the convex side of the Evolute: but if | other lines not perpendicular be drawn upon the same points, provided they be all drawn under the same angle, the effect will still be the same; that is, the oblique lines will all intersect in the curve, and by their intersections form the infinitely small sides of a new curve, to which they would be so many tangents.— Such a curve is a kind of Evolute, and has its radii; but it is an Imperfect one, since the radii are not perpendicular to the sirst curve, or Involute, Hist. de l'Acad. &c, an. 1709.