CHAPTER VI. ANALYTICAL DETERMINATION OF THE CENTRE OF CURVATURE. THEORY OF EVOLUTES AND INVOLUTES. Determination of the Coordinates of the Centre of Curvature. 135. Let a, ẞ, be the coordinates of C, the centre of the osculating circle of a curve AB at the point P. Then, x, y, These two formulæ will enable us to determine a and B for any assigned point of the curve. Suppose s to be the independent variable; then differentiating the equation we have dx2 + dy2 = ds2, and therefore the formulæ for a and B are reduced to Formula for the Coordinates of the Centre of Curvature in terms of Partial Differential Coefficients of u. 136. From Art. (135) we see that and (a − x) dx + (B − y) dy = 0. (1), (a − x) d2x + (B − y) d3y – dx2 – dy2 = 0 . . . . (2). Between these two equations, together with the equation to the curve and its first and second differentials, viz. we may eliminate the six quantities x, y, dx, dy, d'x, d'y. In u 137. Between the equation = 0 and the equations (7) of the preceding Article, we may eliminate x and y: we shall thus obtain an equation between a and ẞ alone, which will be the equation to the geometrical locus of C, the centre of curvature, when the point P of the curve is supposed to be variable in position. The locus of C, for a reason shortly to be explained, is called the evolute of the curve AB, which is itself called the involute. Ex. To find the equation to the locus of the centre of curvature of the ellipse du2 d'u 2 a2 = du du d'u x2 y2 х 29 + b2 du 1. a2 dy b2 dy' de- dz dy de dy de dy o' (5+) - ↓· dx2 dx dx = a2 + a2b2 a2 a2b2 To shew that the Normal at any point of the Involute is a we may 138. By the formulæ (7) of Art. (136) and the equation u = 0, obtain a and ẞ in terms of x; thus a and ẞß, as well as y, are functions of x: hence as x varies, a and ẞ as well as y must simultaneously vary. Differentiating the equation (a - x) dx + (ẞ- y) dy = 0, we shall accordingly obtain (a − x) d2x + (ẞ − y) d3y + da dx + dß dy – dx2 – dy2 = 0, and therefore, by virtue of the equation we have (a − x) d2x + (ẞ − y) d3y – dx2 – dy2 = 0, đa đọc + d dy = 0. This equation shews that the tangent to the involute at the point (x, y) is at right angles to the tangent to the evolute at the corresponding point (a, B). Hence the normal at (x, y), which passes through C, must be a tangent to the evolute at (a, ẞ). Generation of the Involute by the end of a thread unwound from the Evolute. we have and therefore x) ( a − 2) (da – đề) + (B − y) (dB – y) = p dp . . . . (2). Also we know that ... Adding together the squares of (5) and (6), we obtain 2 {(a – x) + (B - y)} (đã + d) = p đó, and therefore, by (1), đã + d3 = dp. Leto denote an arc of the evolute originating at any proposed point and terminating at (a, ẞ): then đã = đã + d = dp, and therefore, the positive or negative sign being chosen accordingly as p decreases or increases with the increase of σ, do + dp = 0, σ + p = c, c being some constant quantity. We proceed now to give the geometrical interpretation of this result. Let C and P be any two corresponding points in the evolute and involute respectively. Let A'CB' be an arc of the evolute. Join CP, which will be the radius of |
