An elementary course of infinitesimal calculus . which it encloses. The formulae (5) of Art. 137 give, for b = 2a, x= — a cos & — 2a cos \6, y= — asm6- 2a sin ^6, or x-a = -2a{\+cos\ff)cos\6, y = -2a{\+cos.^6)sva.^0 (8). If we put & = 16 + IT, it appears that tlie pericycloid, referred to the point (a, 0) aspole, has the equation r = {\-oos6) (9), and is therefore a cardioid. The connection between this result and that of Ex. 1, above,will appear in Art. 168. Ex. 4. The four-cusped hypocycloid. If in Art. 137 (6) we put h = \a, we get x = ^acoa6 + \acosW = acos^0,\ y = lasm6-\ = asm^6]
An elementary course of infinitesimal calculus . which it encloses. The formulae (5) of Art. 137 give, for b = 2a, x= — a cos & — 2a cos \6, y= — asm6- 2a sin ^6, or x-a = -2a{\+cos\ff)cos\6, y = -2a{\+cos.^6)sva.^0 (8). If we put & = 16 + IT, it appears that tlie pericycloid, referred to the point (a, 0) aspole, has the equation r = {\-oos6) (9), and is therefore a cardioid. The connection between this result and that of Ex. 1, above,will appear in Art. 168. Ex. 4. The four-cusped hypocycloid. If in Art. 137 (6) we put h = \a, we get x = ^acoa6 + \acosW = acos^0,\ y = lasm6-\ = asm^6] ^ from which the curve is easily traced. The Cartesian form is !Ki + yS = a3 (11). 138-139] SPECIAL CURVES. 359 This curve is sometimes called the astroid. It possesses theproperty that the length of the tangent intercepted between thecoordinate axes is constant. If, in Fig. 94, P be the tracing-point, TP is the tangent, and it is easily seen that the angleCTP is double the angle AOC. Hence HK= 20T= a. See alsoFig. 134, p. Fig. 94. 139. Superposition of Circular Motions. Epl-cyclics. The cycloidal and trochoidal curves discussed in —138 present themselves in another manner, as thepaths of points whose motion is compounded of two uniformcircular motions. If an arm OQ revolve about a fixed point 0 with constantangular velocity n, its projections on rectangular axes through0 may be taken to be 00=0 cosnt, y = csm nt (1), where c = OQ, provided the origin of t be suitably another arm OQ revolve about 0 with constant angularvelocity n, starting simultaneously with OQ from coincidencewith the axis of x, the projections of OQ will be x = ccosnt, y = csmnt (2), where c = OQ. If we complete the parallelogram OQPQ, 360 INFINITESIMAL CALCULUS. [CH. IX the vector OP will represent the geometric sum of OQ andOQ, and the coordinates of P will be a; = ccosn< + ccos nt, y=camnt-\-csin (3)*,
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