Using Stirling s Interpolation to Find Gauss and Mean Curvature for the Surface
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1 IOSR Joural o Mathematics (IOSR-JM ISSN: - Volume Issue (Nov - Dec PP - wwwiosrjouralsorg Usig Stirlig s Iterolatio to Fid Gauss ad Mea Curvature or the Surace R B M Amer M A Abd El- Mageed (Deartmet o Mathematics ad Ph Egieerig Facult o Egieerig/ Zagazig Uiversit Egt Abstract: I this aer we ca use Stirlig s iterolatio to comute the mea ad gauss curvatures or a surace z = ( at a oit ( Kewords: Dieretial geometr Gauss curvature Mathematica rogram Mea curvature Stirlig s iterolatio I Itroductio Surace characteristics ca be divided ito two tes: - Local characteristics are associated with oits o the surace ad ca be discovered b eamiig the local eighborhoods o oits It is icludig: cotiuit mea curvature gaussia curvature sigularities critical oits: miima maima ad saddle oits - Global characteristics are associated with the surace as a whole ad caot be determied strictl b lookig at local eighborhoods It is icludig: embeddedess orietabilit smmetr icludig eriodicit geus eds total curvature skeletal grahs Deiitio The gaussia curvature [] o a surace at a oit is the roduct o the ricial curvatures K = k k at that oit The taget lae o a oit with ositive gaussia curvature touches the surace at a sigle oit whereas the taget lae o a oit with egative gaussia curvature cuts the surace Deiitio The mea curvature [] hal the sum o the ricial curvatures H = (k + k / ad a oit with zero mea curvature has egative or zero gaussia curvature Suraces with zero mea curvature everwhere are miimal suraces The gaussia ad mea curvature la a ver imortat role i the theor o suraces ad these are deied at each oit o the surace Deiitio The real-valued uctio o two variables z = ( is a surace i [] II The Formulas o Gauss ad Mea curvatures Let k k ( k k deote the two ricile curvature the Gaussia curvature K ad the mea curvature H are deied as K = k k ad H = (k k Formulas or comutig K ad H [] are wwwiosrjouralsorg Page
2 ( Mea curvature = H = Usig Stirlig s Iterolatio to Fid Gauss ad Mea Curvature or the Surace ( ( ( ( Gauss curvature = K = ( Thereore the ricile curvature k k are k H H K k H H K ( We shall deote artial derivatives o uctios b ( III Usig Mathematica Program From Mathematica rogram we ca comute: Ead (a+b rom = This is the rogram to evaluate the ead We use Do to write a rocedural rogram or calculatig the actorial o a iteger b usig a commad ac to calculate the actorial o a ositive iteger ad SetDelaed (:= or we do ot wat to evaluate the right side util ac is called Or use this eas commad: IV Stirlig's Cetral-Dierece Iterolatio Deiitio Stirlig's Cetral-Dierece Iterolatio ormula is based o a diagoal dierece table rather tha a horizotal dierece table wwwiosrjouralsorg Page
3 Usig Stirlig s Iterolatio to Fid Gauss ad Mea Curvature or the Surace Without goig i to the distictios betwee the two tes o dierece tables we ca write dow a iterolatio ormula icludig ol secod diereces that is similar to Newto's iterolatio ormulas Lemma [] Stirlig's iterolatio ormula takes the orm: [htt://eomsrigerde/s/shtm] ( [ ( ( ( h! (! ( ( [ (! Where = ( / h = ( / ( + ( Sice ( ( h ( ( Note that ( = ( = ( = ( Remark For small Stirlig's iterolatio ormula is more eact tha other iterolatio ormulas Deiitio [] The ormula or a cetral dierece ca be m m m / m m m (m = = - ( / / We ca write these euatios as ollows: m m m m / / m m m m m / m / So m ad so o The ( / / ( / / ( / / ( / / / / ( wwwiosrjouralsorg Page From [] k k k k k μ k = = - / / / / (
4 Usig Stirlig s Iterolatio to Fid Gauss ad Mea Curvature or the Surace wwwiosrjouralsorg Page k k k / ( Where [ ] is averagig or mea oerator deied b μ ( We ca write these euatios as ollows: ] [ ] μ[ k k / k k k / k / k / k / k / k k ( The ] [ ] [ ] [ / / ( ] [ ] [ / / / / ( ] [ ] [ / / / / ( ] [ ] [ / / / / ( ] [ ] [ / / / / (
5 Usig Stirlig s Iterolatio to Fid Gauss ad Mea Curvature or the Surace Thus give values o a uctio ( at a iite umber o discrete oits = ad lettig = [( - i / ( i+ - i ] Stirlig's cetral-dierece iterolatio ormula ca be eressed usig uctioal values as ( ( ( (!!!! ( ( ( ( ( ( ( ( (!!! ( ( ( ( ( ( ( (!! From above euatios we ca write this euatio as ( [ ] [ ] [ ] ( [ ] ( ( [ ] ( ( [ ] ( ( ( [ ] ( ( ( [ ] ( ( ( ( ( ( ( ( ( Dieretiate this euatio with resect to ad ut = the d( ( d ( ( ( [ ] [ [ ] [ ] ] The ( ( ( ( ( Sice = +h = ( ( The We get d d d h d d( d h d( d Χ d d Χ Let the iterval legth h = The we ca get at a oit( or the surace z = ( d d ( ( ( ( ( ( ( wwwiosrjouralsorg Page
6 Usig Stirlig s Iterolatio to Fid Gauss ad Mea Curvature or the Surace Dieretiate twice with resect to ad ut = the d ( d ( ( [ ( [ ( ] [ [ ] ] ] The ( ( ( ( ( ( The we ca get at a oit( or the surace z = ( ( ( ( ( ( ( ( ( We ca also id ad at a oit ( or the surace z = ( b usig the same method Sice d( d ( ( d ( d ( ( ( The we get ( ( ( ( ( ( ad ( ( ( ( ( ( To calculate at a oit( or the surace z = ( b Stirlig's cetral-dierece iterolatio ormula which eressed usig uctioal values as ( ( ( h ( ( ( ( (!!! ( ( ( ( ( ( (!! ( ( ( ( ( ( ( (!! ( ( ( ( ( ( ( ( ( (!! Dieretiate this euatio with resect to the ( wwwiosrjouralsorg Page
7 Usig Stirlig s Iterolatio to Fid Gauss ad Mea Curvature or the Surace wwwiosrjouralsorg Page ( ( d d( ( ( ( ( ( ( ( ( ( ( ( h ( ( Let = + h ( = The ( h h
8 Usig Stirlig s Iterolatio to Fid Gauss ad Mea Curvature or the Surace wwwiosrjouralsorg Page ( (
9 Usig Stirlig s Iterolatio to Fid Gauss ad Mea Curvature or the Surace wwwiosrjouralsorg Page ( Dieretiate twice with resect to ad ut = = the ( ( We get: ( ( The
10 Usig Stirlig s Iterolatio to Fid Gauss ad Mea Curvature or the Surace wwwiosrjouralsorg Page ( ( Hece ( ( ( Let h = The we ca get at a oit( or the surace z = ( ( ( ( V Coclusios The most imortat cetral dierece ormula is stirlig iterolatio ad it takes the mea o Gauss orward ad backward ormulae Stirlig 's ormula gives the most accurate result or small ad iterolatio ear the middle o a table Reereces [] K Wolgag Dieretial geometr: curves - suraces maiolds ( America Mathematical Societ ISBN [] O Joh Dieretial geometr ad its alicatios ( Rretice Hall Ic USA [] H Adam M Roald & K Harab Surace curvature estimatio or automatic coloic ol detectio Proceedigs o SPIE Belligham WA Vol [] F B Hildebrad Itroductio to Numerical Aalsis ( New York: Dover Secod editio [] IS Berezi ad NP Zhidkov Comutig methods ( Pergamo (Traslated rom Russia [] S S Sastr Itroductor Methods o Numerical Aalsis (th Ed retice Hall o Idia NewDelhi
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