An elementary course of infinitesimal calculus . nts A, B, A, B areE, F, E, F, respectively. Ex. 3. To find the evolute of a cycloid. At any point P on the cycloid APB (Fig. 125), we have, byArt. 151, Ex. 2, p = 2PI (14). Let the axis AB be produced to D, so that BD = AB; andproduce TI to meet a parallel to BI, drawn through D, in /.If a circle be described on // as diameter, and PI be producedto meet the circumference in P, we have PI=PI, so that P isthe centre of curvature of the cycloid at P. And since the arcPi is equal to the arc TP, and therefore to BI or Dl, thelocus of P is evidently t
An elementary course of infinitesimal calculus . nts A, B, A, B areE, F, E, F, respectively. Ex. 3. To find the evolute of a cycloid. At any point P on the cycloid APB (Fig. 125), we have, byArt. 151, Ex. 2, p = 2PI (14). Let the axis AB be produced to D, so that BD = AB; andproduce TI to meet a parallel to BI, drawn through D, in /.If a circle be described on // as diameter, and PI be producedto meet the circumference in P, we have PI=PI, so that P isthe centre of curvature of the cycloid at P. And since the arcPi is equal to the arc TP, and therefore to BI or Dl, thelocus of P is evidently the cycloid generated by the circle IPI,supposed to roll on the under side of DT, the tracing pointstarting from D. That is, the evolute is a cycloid equal to theoriginal cycloid, and having a cusp at D. It appears, further, from Art. 136 (4), that the cycloidal arcPD is equal to 2/P, or PP. Hence arc i>P + PP = const (15). 424 INFINITESIMAL CALCULUS. [CH. X The lower cycloid iu Fig. 125 is therefore an * involute (Art. 161)of the upper one*.. It may be noted that whenever a curve is defined by arelation between p and ?<fr, say P=f(.f) (16), the evolute is given by P=fW (17), provided that in (17) the origin of -yfr be supposed movedforwards through a right angle. This is seen at once onreference to Fig. 115, p. 401, since OU, the perpendicularfrom the origin on the tangent to the evolute, is equal toPZ, or dp/dyfr, when the symbols refer to the original curve. * This example is interesting historically in connection with the theoryof the cycloidal pendulum. The results are due to Hayghens (1673). 159-160] CURVATURE. 425 Ex. 4. To find the evolute of an epi- or hypo-cycloid. If in Fig. 86, p. 351, a perpendicular p be drawn from 0 toTP, the tangent to the epicycloid at P, we have p = or cos PIO = (a + 26) cos J<^, or ; = (»+2J)cos--^,/r (18). If the origin of i^ correspond to a cusp instead of to a vertex, thecosine of the angle must be replaced by the sine. Hence, for
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