Planar Curves out of Their Curvatures in R

Transcription

1 Planar Curves ou o Their Curvaures in R Tala Alkhouli Alied Science Dearen Aqaba College Al Balqa Alied Universiy Aqaba Jordan doi: 9/esj6vn6 URL:h://dxdoiorg/9/esj6vn6 Absrac This research ais o inroduce soe o he ain ideas o dierenial geoery The research deals wih he ain conces needed o unders his work In his research roeries o curves in R are sudied The research is buil on using he curvaure o a curve in R o derive a araeric orula or he velociy acceleraion Also he geoery o ocal oins has been discussed Exales are buil o suor he ai o his research Keywor: Curvaure Frene Forulas ocal oin lanner curves Inroducion The ain conen o his research is soe dierenial geoery concerning curves in R a brie descriion o curves in R is inroduced The geoery o curves in R is described using Frene orulas The conce o a ocal oin is a base wihin his research so deiniions heores concerning ocal oins are included As his research is abou geoery o curves he curvaure o a curve in R is sudied in he ield o ocal oins Dierenial geoery is a aheaical ield which allows he sudy o geoerical conces using calculusthus calculus is used o exlain how o exrac lanar curves ou o heir curvaures an idea which suors he conen o his research Exales are buil o suor he ai o his research Oher geoerical conces are included R Curves in Deiniion A curve C in is an oen inerval Thus i is a araeer o where R is a diereniable uncion hen we wrie as : R are diereniable real valued uncions deined on where

2 We say ha is a araerizaion o C or C is he race o on Deiniion A curve C in R wih araerizaion : R is regular rovided ha or all I or soe hen is a singular oin o C Deiniion Le : R be a curve in R Le : J be a diereniable real valued uncion where J is also an inerval Then w : J R is a rearaerizaion o by Deiniion The rearaerizaion w o is orienaion reserving i on J orienaion reversing i on J ab deine he arclengh along C by Now we ake where s u du a is he nor o usually called he seed o I C is a regular curve hen or all so s is a sricly increasing uncion o which has an inverse Tha is Equaion can be solved or s hen C has a rearaerizaion by arclengh Proosiion 5 Le w be an arclengh rearaerizaion o Then w has uni seed Proo: Observe ha w is deined by w Then diereniae wih resec o s o ge dw d d d d d d Rearaerizing a regular curve by acrlengh also called uni seed rearaerizaion siliies calculaions regarding geoery o curves R Curvaure o a curve in Now ake w as a uni seed curve hen w w For w = w w w w I s is he uni angen o w a s hen w Also w Deiniion The curvaure o a uni seed curve w where is he uni angen o w Now le be he uni noral o w a s is he uncion

3 hen The orhonoral se o he vecors is called Frene rae So can be wrien in ers o In ac i a b hen b by Equaion a Hence Equaions are called Frene orulas Now le be an arbirary regular curve wih araeer uni seed rearaerizaion w Le Theore Frene orulas o are 5 Proo: I is clear ha here is a corresondence beween oins on w in a way ha s such ha w w Thus Equaion ilies ha d d d Also Equaion ilies ha d T d d Theore The velociy acceleraion vecors o a are resecively 6 7 Proo: The uni angen o a is so so Equaion 6 is derived w w

4 Now diereniae Equaion 6 wih resec o hen use Equaion o ge Equaion 7 Theore Le be an arbirary regular curve Then he curvaure o is given by 8 Proo: By Equaions 6 7 we have v v Thus Hence Equaion 8 ollows Deiniion 5 The oal curvaure o a regular curve on is d R The ocal curve o a curve in Deiniion The ocal curve o a regular curve is he curve g Siilarly he ocal oin wih base is he oin g Now le Λ be he disance uncion whose doain is he curve C wih araerizaion Theore The oin is a ocal oin o C wih base i Proo: Recall ha Thus And I is a ocal oin o C wih base hen Thus 5

5 6 Also Conversely he equaion ilies ha Now u in he equaion o ge which is equivalen o or Thus So ie is a ocal oin o C wih base Exale Le Then So Thus By solving he equaion or we ge Thus

6 g Now a b wih a b So b a b Equaing he las wo equaions o zero we ge b a Exale Consider he curve deined by : 9cos cos cos sin sin9sin sin cos cossin Then 9 8cos sinco 9 8cos cos sin sin sinco So 9 8cos 9 8cos Thus 9 8cos By solving he equaion sin sin co 9 8cos 9 8cos or we ge cos sin Thus g 9cos cos sin sin9sin cos cos sin Le a b Then 9cos cos cos sin sin a 9sin sin cos cossin b So 9 8co sin asin bco 7

7 9 8cos 9cos acos bsin 8sin sin asin bco Equaing he las wo equaions o zero we ge sin asin bcos 9cos acos bsin Solving he las wo equaions ogeher we ge a 9cos cos sin sin b 9sin cos cos sin Planar curves ou o heir curvaures Le be a lanar curve wih araeer s he arclengh Le be he sloe angle o he angen line a Then he uni angen o a s is deined by cos sin d d Now k s Thus s k 9 Now i x y hen x s cos y s sin Exale Le k wih Then by Equaion 9 s k s By Equaion we have x cos s sin s c Bu x so x sin s Also by Equaion we have 8

8 y sin s cos s c Bu y so y cos s So sin s cos s which is a circle o radius Exale Le k s wih Then by Equaion 9 s s k s s an Now by Equaion we have x cosan sinh s x sinh s c Bu x so s Also by Equaion we have y sinan s s c s Bu y so y s So sinh s s Reerences: Al-Banawi K Generaing Fraes Noral Holonoy o Transnoral Subaniol in Euclidean Saces PhD hesis Universiy o Lee UK Al-Banawi K Geoery o Ovals in Ters o he Suor Funcion Journal o Alied Sciences 88:8-86 Al-Banawi K Soe Noes on Classical Dierenial Geoery Dearen o Maheaics Saisics Muah Universiy Course Noe Eggleson H Convexiy Cabridge Tracs in Maheaics 7 Cabridge Universiy Press 958 9

9 5 Hsiung C A Firs Course in Dierenial Geoery John Wiley Sons 98 6 O Neill B Eleenary Dierenial Geoery nd Ediion Acadeic Press London Sruik D Lecures on Classical Dierenial Geoery Addison- Wesley New York 95