HYPOCYCLOID ViflTH FOUR CUSf S 11.1

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1 Y 11.1 E i p q c n o uo l o ta r r i d o i nn, j, r = c s 1 E 1 i r q. cn e u 2 o c o t t r i d n - o i x g n ( + y = C S - y* G s ) ( )! & 2,, 1 A b1 An o A e. B x g r t = 3 4n lx B w 5 d e i e s e n A l B, / 1 A o 1 o l r = & f. n o e 4 e o 2 p F 1 i 1 g- C Y C l O 11.5 E i p q f n u o r r ty m m i : e o t n [ C= CE L - ( s + + i ) n 1 y = - C ( O 1 S # ) 1 A o 1 o r = 3 f. n r e = 6 e c h 2 1 A l 1 o o r e. = 8f n c r n 7 e c g h t h T i c dh s bu pei F o y r c os o r n v i i c f e r nr d c t i i l b u r x o l xl o li n is g n. F 1 g i 1 g - HYPOCYCLOID ViflTH FOUR CUSf S 1 E 1 i r q. cn e u 8 o c o t t r i d n o i g n % + y 2 Z Z / Z 2 f 3 Z l E 1 i p q. f n u 9 o r r t m m i : e o t n x y = = C O S 3 9 s 0 i n z A b b r c o = & y e u u r n 2 v d e e d A l o e r ec f = n6 c nu t gr i tv r h e e T i c dh s bu pei P o y r c os o r n v i i c f e r nr d c t i l b u F 1 i 1 g - u i r o / t s i t o o n c4 h no rl f i e sf l r. i s d c d i l e u e s 40

2 . SPECIAL PLANE CURVES 41 CARDIOID Eqution: r = (1 + COS 0) Are bounded by curve = $XL~ Arc length of curve = 8 This is the curve described by point P of circle of rdius s it rolls on the outside of fixed circle of rdius. The curve is lso specil cse of the limcon of Pscl [sec Fig CATEIVARY Eqution: Y z : (&/ + e-x/) = coshs This is the eurve in which hevy uniform chm would hng if suspended verticlly from fixed points A nd. B. Fig THREEdEAVED ROSE Eqution: r = COS 39 Y The eqution T = sin 3e is similr curve obtined by rotting the curve of Fig counterclockwise through 30 or ~-16 rdins. X, In generl v = cs ne or r = sinne hs n leves if / n is odd.,/ / +, Fig FOUR-LEAVED ROSE Eqution: r = COS 20 The eqution r = sin 26 is similr curve obtined by rotting the curve of Fig counterclockwise through 45O or 7714 rdins. In generl n is even. y = COS ne or r = sin ne hs 2n leves if Fig. 11-7

3 42 SPECIAL PLANE CURVES Prmetric equtions: X = ( + b) COS e - b COS Y = ( + b) sine - b sin This is the curve described by point P on circle of rdius b s it rolls on the outside of circle of rdius. The crdioid [Fig is specil cse of n epicycloid. Fig GENERA& HYPOCYCLOID Prmetric equtions: z = ( - b) + b COS Il = (- b) sin + - b sin This is the curve described by point P on circle of rdius b s it rolls on the inside of circle of rdius. If b = /4, the curve is tht of Fig Fig TROCHU#D Prmetric equtions: x - 1 sin 4 v = -bcos+ This is the curve described by point P t distnce b from the tenter of circle of rdius s the circle rolls on the z xis. If 1 <, the curve is s shown in Fig nd is clled cz&te c~czos. If b >, the curve is s shown in Fig. ll-ll nd is clled prozte c&oti. If 1 =, the curve is the cycloid of Fig Fig Fig. ll-ll

4 SPECIAL PLANE CURVES 43 TRACTRIX Prmetric equtions: x = u(ln cet +$ - COS #) y = sin+ This is the curve described by endpoint P of tut string PQ of length s the other end Q is moved long the x xis. Fig WITCH OF AGNES Eqution in rectngulr coordintes: u = 8~x3 x Prmetric equtions: x = 2 cet e y = (1 - cos2e) Andy In Fig the vrible line OA intersects y = 2 nd the circle of rdius with center (0,~) t A respectively. Any point P on the witch is locted oy constructing lines prllel to the x nd y xes through B nd A respectively nd determining the point P of intersection. -q-+jqx l Fig FOLIUM OF DESCARTRS Eqution in rectngulr coordintes: Y x3 + y3 = 3xy Prmetric equtions: 1 x=m 3t 3t2 y = l+@ Are of loop = $2 Eqution of symptote: x+y+u Z 0 Fig INVOLUTE OF A CIRCLE il.28 Prmetric equtions: x = ~(COS + sin $J) I y = (sin cs +) This is the curve described by the endpoint P of string s it unwinds from circle of rdius while held tut. jy!/--+$$x. I Fig. Il-15

5 44 S P P C L E U A C R N I V E A EVOWTE OF Aff ELLIPSE E i r q cn e u o c o t t r i d n o i g (xy 3 + (bvp3 = tu3 - by P e q r u m t e i t o c = ( - b COS3 z 8 C s z G ) b = ( - b y 2 2 ) 1 s 6 i n s T c i t he u s o ht i n r tf the s eov v o he lre e e lml i o ppi x + y = 1 e s z d / i hlf 1 n bo i 1s 2 s wg -h n. 1e 6d F 1 i 1 g - O OF CASSINI V A L S 1 1 P e 1 of + q l~ i 2 u3 = b O e 4 W r S t i o T i t c hd _--- s h ub pie P e s y r t s ost p u v ho icih dr c e f nrte ti o h f t rp t is ws d i o b [ 2 d i c p i b s o 2 s n. r t s t t ] n c n T c i ih Fu 1 s s o n Fe i r1 1 r i g v1 -b < c og. 1 e > s - 1 rc r. 1 7 eo 8 sr I b = u t cf i, Zh u s [ e1e r F m1 v i -k e g 1c. 1 ++Y!--- P X F 1 i 1 g -. 1 F 17 i 1 g -. LIMACON OF PASCAL P e or = qb l u+ r tc i o o s L O b l ej Q e o i 0 t ot rp n Q io n c io e o d n y i g i p f ii t r in 0 Tn h c nt s. mhg r t c i t hl u o s ph oe P rsf 1t oe Pc = vub 1 h i Q u. e c n s h t t s T c i ih F u 1 s os n Fe i 1r 1 r i g 1v -b > c og. b s < -e 1 rc r. 2 I9 eo1 = t 0 f sr, h c i c u [ s 1 r F r 1 v id - e i g 4 o. 1 i. d - F 1 i 1 g -. 1 F 1 9 i 1 g-

6 SPECIAL PLANE CURVES 45 C OF L l BS IS OO C Eqution in rectngulr coordintes: x 3 y ZZZ x Prmetric equtions: i x = 2 sinz t 2 sin3 e?4 =- COS e This is the curve described by point P such tht the distnce OP = distnce RS. It is used in the problem of duplicution of cube, i.e. finding the side of cube which hs twice the volume of given cube. Fig SPfRAL OF ARCHIMEDES Polr eqution: Y = 6 Y Fig

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