Y 11.1 E i p q c n o uo l o ta r r i d o i nn, j, r = c 2 2 2 0 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 3 3 1 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 11.10 A b b r c o = & y e u u r n 2 v d e e d 11.11 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
. SPECIAL PLANE CURVES 41 CARDIOID 11.12 Eqution: r = (1 + COS 0) 11.13 Are bounded by curve = $XL~ 11.14 Arc length of curve = 8 This is the curve described by point P of circle of rdius s it rolls on the outside of fixed circle of rdius. The curve is lso specil cse of the limcon of Pscl [sec 11.321. Fig. 11-4 CATEIVARY 11.15 Eqution: Y z : (&/ + e-x/) = coshs This is the eurve in which hevy uniform chm would hng if suspended verticlly from fixed points A nd. B. Fig. 11-5 THREEdEAVED ROSE 11.16 Eqution: r = COS 39 Y The eqution T = sin 3e is similr curve obtined by rotting the curve of Fig. 11-6 counterclockwise through 30 or ~-16 rdins. X, In generl v = cs ne or r = sinne hs n leves if / n is odd.,/ / +, Fig. 11-6 FOUR-LEAVED ROSE 11.17 Eqution: r = COS 20 The eqution r = sin 26 is similr curve obtined by rotting the curve of Fig. 11-7 counterclockwise through 45O or 7714 rdins. In generl n is even. y = COS ne or r = sin ne hs 2n leves if Fig. 11-7
42 SPECIAL PLANE CURVES 11.18 Prmetric equtions: X = ( + b) COS e - b COS Y = ( + b) sine - b sin This is the curve described by point P on circle of rdius b s it rolls on the outside of circle of rdius. The crdioid [Fig. 11-41 is specil cse of n epicycloid. Fig. 11-8 GENERA& HYPOCYCLOID 11.19 Prmetric equtions: z = ( - b) COS @ + b COS Il = (- b) sin + - b sin This is the curve described by point P on circle of rdius b s it rolls on the inside of circle of rdius. If b = /4, the curve is tht of Fig. 11-3. Fig. 11-9 TROCHU#D 11.20 Prmetric equtions: x = @ - 1 sin 4 v = -bcos+ This is the curve described by point P t distnce b from the tenter of circle of rdius s the circle rolls on the z xis. If 1 <, the curve is s shown in Fig. 11-10 nd is clled cz&te c~czos. If b >, the curve is s shown in Fig. ll-ll nd is clled prozte c&oti. If 1 =, the curve is the cycloid of Fig. 11-2. Fig. 11-10 Fig. ll-ll
SPECIAL PLANE CURVES 43 TRACTRIX 11.21 Prmetric equtions: x = u(ln cet +$ - 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. 11-12 WITCH OF AGNES1 11.22 Eqution in rectngulr coordintes: u = 8~x3 x2 + 42 11.23 Prmetric equtions: x = 2 cet e y = (1 - cos2e) Andy In Fig. 11-13 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. 11-13 FOLIUM OF DESCARTRS 11.24 Eqution in rectngulr coordintes: Y x3 + y3 = 3xy 11.25 Prmetric equtions: 1 x=m 3t 3t2 y = l+@ 1 11.26 11.27 Are of loop = $2 Eqution of symptote: x+y+u Z 0 Fig. 11-14 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
44 S P P C L E U A C R N I V E A EVOWTE OF Aff ELLIPSE 11.29 E i r q cn e u o c o t t r i d n o i g (xy 3 + (bvp3 = tu3 - by3 11.30 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 + q4. - 2 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 11.32 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-
SPECIAL PLANE CURVES 45 C OF L l BS IS OO C 11.33 Eqution in rectngulr coordintes: x 3 y ZZZ 2 2 - x 11.34 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. 11-21 SPfRAL OF ARCHIMEDES 11.35 Polr eqution: Y = 6 Y Fig. 11-22