An elementary course of infinitesimal calculus . esponding position of thepoint of contact of the rolling circle. Cf. Art. 164. 137. Epicycloids and Hypocycloids. The path traced out by a point on the circumference of acircle which rolls in contact with a fixed circle is called an epicycloid or a hypocycloid according as the rolling circleis outside or inside the fixed circle*. Those epicycloids inwhich the rolling circle surrounds the fixed circle may berefen-ed to, when a distinction is desired, as pericycloids. Let 0 be the centre of the fixed circle, G that of therolling circle in any posi
An elementary course of infinitesimal calculus . esponding position of thepoint of contact of the rolling circle. Cf. Art. 164. 137. Epicycloids and Hypocycloids. The path traced out by a point on the circumference of acircle which rolls in contact with a fixed circle is called an epicycloid or a hypocycloid according as the rolling circleis outside or inside the fixed circle*. Those epicycloids inwhich the rolling circle surrounds the fixed circle may berefen-ed to, when a distinction is desired, as pericycloids. Let 0 be the centre of the fixed circle, G that of therolling circle in any position, I the point of contact, P thetracing point; and suppose that, initially, the other extremityP of the diameter PGP was in contact with A. We take * This is the definition as improved by Proctor in his Treatise on theCycloid, etc. (1878). 136-137] SPECIAL CURVES. 351 as our standard case that in which each circle is external tothe other. Let OA = a, GP = b, /.IOA = e, /iIGP = 4>. The inclination of GF to OA will be ^ + <^. Hence if we. Fig. 86. take 0 as origin of rectangular coordinates, and OA as axisof x, we find, by orthogonal projections, that the coordinatesof P are or, since a; = {a + b) cos 6 + b cos {0 + <^),y = {a + b) sinO + b sin (d + = (a — b) cos 0 — b cos —j— i ,(5). y = (a~b)sin0 + b sin —=— 6 The verification is left to the reader; see Fig. 87. In thehypocycloids we have a >
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