6 Supeellipse (Lmé cuve) 6. Equtios of supeellipse A supeellipse (hoizotlly log) is epessed s follows. Implicit Equtio y + b 0<b, > 0 (.) Eplicit Equtio y b - 0<b, > 0 (.') Whe 3, b, the supeellipses fo 9.9,, d /3 e dw s follows. Fig These e clled ellipse whe, e clled dimod whe, d e clled steoid whe /3. These e kow well. pmetic epesettio of ellipse I ode to sk fo the e d the c legth of supe-ellipse, it is ecessy to clculus the equtios. Howeve, it is difficult fo (.) d (.'). The, i ode to mke these esy, pmetic epesettio of (.) is ofte used. As f s the st qudt, (.) is s follows. y 0 + b 0yb Let us compe this with the followig tigoometic equtio. si + cos The, we obti i.e. si, y b cos - -
si, y bcos The vible d the domi e chged s follows. : 0 : 0/ By this epesettio, the e d the c legth of supeellipse c be clculted clockwise fom the y-is. cf. If we epeset this s usul s follows, cos, y bsi the e d the c legth of supeellipse c be clculted couteclockwise fom the -is. Howeve, fom this epesettio, we cot obti the Legede foms of elliptic itegls - -
6. Ae of supeellipse Fomul 6.. Whe,,b ( b ) e positive umbes espectively d ( z) is the gmm fuctio, the e S of the ellipse of degee is give by the followig epessio. Poof S 4b + + I ode to obti the e of supeellipse, we itegte with the followig equtio i the st qudt, d should just multiply it by 4. Tht is, y b - S 4b 0 - d 0<b, > 0 Howeve, it is kow tht this itegtio cot be epessed with elemety fuctios. The, we use the pmetic epesettio metioed i the pevious sectio. Tht is, si, y bcos The vible d the domi e chged s follows. Ad d is : 0 : 0/ The, d si - cos d S 4 0 yd 0 bcos ( si -cos)d -cos b si + d 0 Accodig to " 岩波数学公式 Ⅰ" p 43, 0 Usig this, si cos d -cos si 0 + +, -+,, > - + d ++, + - 3 -
Substitutig this fo the vove, S b 4 + + + + + b Quduplig both sides, we obti the desied epessio. Note Especilly whe b, 4 + / + + + + + is the e of the uit cicle + y > 0, d should be clled costt of the cicle of degee. Tht is, the e of ellipse of degee is give by b, whee is the mjo is d b is the mio is. I fct, whe, 4 + + 4! 4 This is the e of the cicle + y, d the e of ellipse of degee is give by b. Emple: Ae of ellipse of degee.3, 3, b As the esult of clcultig this e i umeicl itegl d the fomul, both wee completely cosistet. - 4 -
6.3 A pt of e of supeellipse I this sectio, we clculte the e of the light-blue potio of the followig figue. Fomul 6.3. Whe,,b ( b ) e positive umbes espectively, the e s( ) degee i the st qudt is give by the followig epessio. s( ) b / fom 0 to of the ellipse of (-) + (.) + Poof The e s( ) fom 0 to of the supeellipse i the st qudt is give by the followig itegl. b 0 - s( ) d 0< This itegl is ot epessed with elemety fuctios ecept fo specil cse. So, we ty the temwise itegtio. Applyig geelized biomil theoem to this itegd, - ( ) / - Itegtig the ight side with espect to fom 0 to tem by tem, - 0 d ( ) Multiplyig this by b, we obti (.). / - / 0 (-) + + d Emple: Ae of the ellipse of degee.3, 3, b o 0-5 -
We clculte the e of the light-blue potio of the bove figue. As the esult of clcultig by umeicl itegl d (.), both wee completely cosistet. By-poducts Especilly itegtig this to, b 0 - d b The, the e S of the supeellipse is S 4b Theefoe lso, + / / (-) + + b / (-) + (-) (.) + / + / (-) (.3) + - 6 -
6.4 Ac legth of supeellipse 6.4. Ac legth of oblog supeellipse I this sub-sectio, we clculte the legth bp of the followig oblog supeellipse. y b - 0<b, > 0 (.0) Fomul 6.4. Let,,b ( b ) e positive umbes, be umbe s.t. 0< z s be Pochhmme's symbol. The, the c legth l( ) d () l( ) / / ( -) -+s ( -) - ( -) / -s ( ) - ( -s) s s! s s! +s +s b - + - fom 0 to of (.0) is give by the followigs. ( -) +s+ (.) b ( -) +s+ ( -) +s+ (.') b ( -) +s+ -s ( ) b ( ) ( -s ) +s ( - )( -s ) +s+ (.") - ( -s ) +s+ Poof Legth l( ) of the cuve y f( ) o ple is give by the followig fomul. l( ) dy + d d Diffeetitig both sides of (.0) with espect to, dy b d - - b / - / - - - - - b - - - - - - 7 -
The, l( ) 0 / + b - / ( -) d () This itegl is ot epessed with elemety fuctios ecept fo specil cse. So, we epd this itegd to seies. b - - / 0< + Accodig to gelized biomil thoem, The, b + l( ) / - / / ( -) b 0 b 0 < / - / / - / b Sice this itegl is lso o-elemety fuctio, we ty the temwise itegtio. + + + - s+0 0 s ( -) / - / ( -) ( -) d () Diffeetitig both sides of this with espect to d dividig it by fctoil oe by oe, ( -) ( -) 3 ( -) m Multiplyig both sides by m, m ( -) m Replcig with /, / - / 0 + + 3 + 4 3 +! 0 + 3 + 6 + 0 3 +! m s+m- m- m+s s+m- m- Itegtig both sides with espect to fom 0 to, i.e. 0 / - / m d 0 / m - / d s+m- m- 0 ( m+s) m-+s m- m+s ( m+s) d m+s+ m+s+ s+ s+ s s s+m- m- s - 8 -
Sice m my be el umbe, eplcig this with 0 / - / ( -) Substitutig this fo (), we obti Net, The l( ) / m-+s m- / d b m-+s s ( -) -+s ( -) - ( ) ( -), ( -) -+s - ( ) - ( ) ( -) -+s - ( ) - ( ) - -+s ( -) - ( m+s) ( m) ( s+) ( -) s! s +s - +s - +s ( - ) +s+ ( -) +s+ ( - ) +s+ ( -) +s+ ( -) +s+ (.) b ( -) +s+ ( m) s s! Substitutig thes fo (.), we obti (.'). Futhemoe, egig (.') to the digol seies, we obti (."). Emple: Ac legth of the ellipse of degee.5, 3, b o 0.5449 Whe 3, b, 5/, the lgest clculble i the fomul 6.4. is s follows. If the c legth o :0.5449 e clculted by umeicl itegtio d (.'), it is s follows. As the esult of clcultig to 30 espectively, both wee cosistet util 7 digits below the deciml poit. - 9 -
6.4. Ac legth of logwise supeellipse I Fomul 6.4., we cot clculte the c legth P of oblog supeellipse. Howeve, this poblem is esily solvble. Tht is, let us eplce d y i (.0), s follows. y - b 0<b, > 0 (.0) The, the legth of P of oblog supeellipse is cosistet with the legth of P of logwise supeellipse. Ad i ode to obti this, we should just eplce d b i Fomul 6.4.. Fomul 6.4. Let,,b ( b ) e positive umbes, be umbe s.t. 0< b () z s be Pochhmme's symbol. The, the c legth l( ) l( ) / / ( -) -+s ( -) - ( -) / -s ( ) - ( -s) s s! s s! - + b - d fom 0 to of (.0) is give by the followigs. ( -) +s+ b +s b +s ( -) +s+ (.) ( -) +s+ (.') ( -) +s+ -s ( ) ( ) b ( -s ) +s ( - )( -s ) +s+ (.") - ( -s ) +s+ - 0 -
Emple: Ac legth of the ellipse of degee.5, 3, b o 3.5449 Whe 3, b, 5/, the lgest clculble i the fomul 6.4. is s follows. The c legth of (.0) o :0.947 is cosistet with the c legth of (.0) o :3.5449. If the c legth o :0.947 e clculted by umeicl itegl d (."), it is s follows. As the esult of clcultig to 5, both wee cosistet util 9 digits below the deciml poit. Note I the logwise supeellipse, the covegece speed of the double seies (.) (o (.')) is slow. It is bout /00 of the covegece speed of the digol seies (."). - -
6.5 Peiphel legth of supeellipse Fomul 6.5. Whe,,b ( b ) e positive umbes espectively, the peiphel legth L of the ellipse of degee is give by the followig epessios. L 4 4 4 Poof Whee, / / ( -) -+s ( -) +s+ + b - B - ( -) +s+ b - A - ( -) ( -) +s+ - ( -) / s s! ( - )( -s) -s Substitutig to s follows. l b - A - s s! b ( -s) -( -s) A - ( - )( -s ) + s+ b - A + A b - + Net, substitutig b fom to s follows. l bb / ( -) +s+ + b - B - ( -) +s+ ( -) +s+ ( - )( -s ) +s+ + ( -s) b -( ) -, B + b - ( -) - + b / -s B - ( - )( -s ) + s+ A fo Fomul 6.4., we obti the c legth fom 0 ( -) -+s - - bb ( -) -+s ( -) - b A - ( -) +s+ ( -) +s+ fo Fomul 6.4., we obti the c legth bb - ( -) +s+ b ( -) +s+ Addig both, we obti the c legth fom 0 to. Quduplig this, we obti (.). Ad ewitig this with Pochhmme's symbol, we obti (.'). Futhemoe, egig this to the digol seies, we obti (."). Emple: Peiphel legth of the ellipse of degee.5, 3, b We clculte this peiphel legth by umeicl itegl d by the bove fomul espectively. Sice the umeicl itegl c ot be ccutely clculted, We dd two itegtio vlues i the pevious sectio, d quduple it. Fomul (.") is the oly choice fo the covegece speed. As the esult of clcultig to 45, both wee cosistet util 9 digits below the deciml poit. (.) (.') (.") - -
0..05 06.0.0 eewed Alie's Mthemtics Ko. Koo - 3 -