Plane and solid analytic geometry; an elementary textbook . value in the second equation and substituting in thefirst, we have x = a vers-1 (-) — V2 ay — y2. But this single equation is not so convenient to use asthe pair of equations from which it was obtained. Thesetwo equations, containing a third variable, are equivalentto the single equation from which 6 has been locus consists of an infinite number of branches,similar to the one shown in Fig. 103, extending both tothe right and to the left of the origin. 106. The hypocycloid.—The path described by a point onthe circumferen
Plane and solid analytic geometry; an elementary textbook . value in the second equation and substituting in thefirst, we have x = a vers-1 (-) — V2 ay — y2. But this single equation is not so convenient to use asthe pair of equations from which it was obtained. Thesetwo equations, containing a third variable, are equivalentto the single equation from which 6 has been locus consists of an infinite number of branches,similar to the one shown in Fig. 103, extending both tothe right and to the left of the origin. 106. The hypocycloid.—The path described by a point onthe circumference of a circle which rolls on the inside of afixed circle is called a hypocycloid. Let a be the radius ofthe fixed circle,and b the radiusof the rolling cir-cle. Let P bethe fixed point onthe rolling the centre0 of the fixedcircle as the ori-gin of coordinatesand let the X-axispass through A,one of the pointswhere P coin- FlG- 104- cides with the fixed circle. Consider the rectangularcoordinates (x, y) of any position of P. Drop perpen-. a x 202 ANALYTIC GEOMETRY [Ch. XV, § 106 diculars from C and P to the X-axis, and through P drawLR parallel to that axis. The radius OK of the fixedcircle, drawn through the point of contact K, passesthrough the centre C of the rolling circle. Let Z A0K= $,and ZPCK= 6. Then, since the arcs AK and PK are equal, acf) = bd, or 6 = - .o Now x = 031= ON- PL,y = MP = NC-LC. But ON =00cos = (a - 5) cos = (a-b) sin ,and LC= b sin (0 — <£)= 5 sin (—-jr—- )<£• Then x = (a — 5) cos $ + 6 cos f ——— ] <£,y = (a — b~) sin $ — b sin ( ~~ ) . As in the cycloid, these two equations, containing athird variable , may be used in place of a single equa-tion in x and y to represent the curve. The most important special case is the four-cusped hypo-cycloid, in which a = 4 b. The equations here reduce to x = | a cos
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