THE INVOLUTE-EVOLUTE OFFSETS OF RULED SURFACES *

Transcription

1 Iranian Journal of Scinc & Tchnology, Tranacion A, Vol, No A Prind in h Ilamic Rpublic of Iran, 009 Shiraz Univriy THE INVOLUTE-EVOLUTE OFFSETS OF RULED SURFACES E KASAP, S YUCE AND N KURUOGLU Ondokuz Mayi Univriy, Scinc and Ar Faculy, Dparmn of Mahmaic, Kurupli 559, Samun, Turky, kaap@omudur Yıldız Tchnical Univriy, Faculy of Ar and Scinc, Dparmn of Mahmaic, Enlr, 40, Ianbul, Turky, ayuc@yildizdur Univriy of Bahchir, Faculy of Ar and Scinc, Dparmn of Mahmaic and Compur Scinc, Bika, 400, Ianbul, Turky, kuruoglu@bahchirdur Abrac In hi udy, a gnralizaion of h hory of involu-volu curv i prnd for ruld urfac bad on lin gomry Uing lin inad of poin, wo ruld urfac which ar off in h n of involu-volu ar dfind Morovr, h found rul ar clarifid uing compur-aidd xampl Kyword Ruld Surfac, involu-volu, diffrnial gomry INTRODUCTION A urfac i aid o b "ruld" if i i gnrad by moving a raigh lin coninuouly in Euclidan pac E Ruld urfac ar on of h impl objc in gomric modling On imporan fac abou ruld urfac i ha hy can b gnrad by raigh lin On would nvr know hi from looking a h urfac or i uual uaion in rm of x, y and z coordina, bu ruld urfac can all b rwrin o highligh h gnraing lin A pracical applicaion of ruld urfac i ha hy ar ud in civil nginring Sinc building marial uch a wood ar raigh, hy can b hough of a raigh lin Th rul i ha if nginr ar planning o conruc omhing wih curvaur, hy can u a ruld urfac inc raigh lin xi vrywhr on h urfac Sinc h dicovry of prconrain concr in 90, archicural conrucion in h hap of ruld urfac hav bn innumrabl, including war-owr, chimny-pic, roof, and piral airca Ero Saarinn (90-96) ud ruld urfac in hi building a Yal and MIT, bu h man who ha ud ruld urfac mor han anyon l i dignr, archic and buildr Félix Candla who mak xniv u of cylindr and h mo familiar ruld urfac In ohr ca, uch a h Chapl a Loma d Curnavaca, Morlo, Mxico, h nir building i a giganic hyprbolic paraboloid A rauran a Xochimilco, D F, i mad of four inrcing hyprbolic paraboloid giving h imprion of an immn callop-dgd hll Thr ar rcn work abou ruld urfac: Alanr udid h im-lik hyprruld urfac in h Minkowki 4-pac [] Toun and Gungor dcribd a im-lik complmnary ruld urfac in h Minkowki n-pac Alo, hy invigad rlaion conncd wih an aympoic and angnial bundl of h im-lik complmnary ruld urfac [] Karadağ and Klş udid h ingral invarian of h ruld urfac in h lin pac corrponding o h clod phrical curv by h ranfrnc principl of E Sudy; for hi hy ud h ara vcor of h clod dual phrical curv [] Rcivd by h dior Fbruary 7, 008 and in final rvid form Dcmbr, 009 Corrponding auhor

2 96 S Yuc / al Huyghn fir inroducd h concp of h involu and h volu in 97 [4] A pair of curv ar aid o b involu-volu ma in Euclidan pac E if hr xi a on-o-on corrpondnc bwn hir poin uch ha on angn and h ohr principal normal ar linar dpndn a hir corrponding poin Mhod for h gnraion of paralll off for a crain cla of urfac hav bn dvlopd by Farouki [5-6] Uing h am chniu, h hory of Brrand curv ha bn dvlopd for h ruld and dvlopabl urfac by Ravani and Ku [7] In hi papr, involu-volu off of ruld urfac ar conidrd Uing lin gomry, i i hown ha a hory imilar o ha of involu-volu curv can b dvlopd for a ruld urfac Th condiion for wo ruld urfac o b involu-volu ma i dvlopd and h rul ar clarifid uing compur-aidd xampl PRELIMINARIES A ruld urfac in -dimnional Euclidan pac E i a urfac wp ou by a raigh lin paralll o along a curv α and ha h paramric rprnaion (, v) α( v(, Th curv α α () i calld h ba curv and h variou poiion of h gnraing lin () ar calld h ruling of h urfac Th curv, which i drawn by ( on h uni phr S i calld h phrical indicarix curv and i alo calld h phrical indicarix vcor of Th uni normal of along a gnral gnraor l ( 0, v) of h ruld urfac approach a limiing dircion a v infinily dcra Thi dircion i calld h aympoic normal dircion and i dfind a g ( ), d d Th poin a which h uni normal of i prpndicular o g i calld h ricion poin (or cnral poin) on l and h curv drawn by h poin i calld h ricion curv of For h ricion curv of, w hav α, c( α( (, Th dircion of h uni normal a a ricion poin i calld h cnral normal of and i givn by Iranian Journal of Scinc & Tchnology, Tran A, Volum, Numbr A Spring 009 Thu, w hav h orhonormal ym {,, g} Thi ym i calld h godic Frn rihdron of For h godic Frn vcor, and g, w can wri g, () g whr and ar h arc-paramr of h phrical indicarix curv () and h godic curvaur of () wih rpc o h uni phr S, rpcivly, [7] In hi papr, h ricion curv of h ruld urfac will b akn a h ba curv In hi ca,

3 Th involu-volu off of ruld urfac 97 for h paramrizaion of, w can wri (, v) c( v( INVOLUTE-EVOLUTE OFFSETS OF RULED SURFACES IN L and b wo ruld urfac in E i aid o b an involu off of (or i aid o b an volu off of ), if hr xi a on-o-on corrpondnc bwn hir ruling uch ha h cnral normal of and h phrical indicarix vcor of ar linarly dpndn a h ricion poin of hir corrponding ruling Th ba ruld urfac, (, v), can b xprd a (, v) c( v(, whr c i i ricion curv and i h arc lngh along c Th uaion of h off urfac, in rm of i ba urfac, can b wrin a v v R v (,) c () () [() c ()()] (), () whr R i h dianc bwn h corrponding ricion poin of and Morovr, inc h ricion curv of i i ba curv, w hav c, R (), If, and g ar h godic Frn vcor of, hn h godic Frn vcor of h volu off of ar givn by E g 0 in θ coθ whr i h angl bwn and g Now, w can giv h following horm for : coθ, () in θ g Thorm L b h volu off of If i conan, hn i conan and alo for h convr, if 0 hn h convr i ru Proof: From h dfiniion of Bcau of uaion (), w obain Th la uaion impli ha, w g g Spring 009 Iranian Journal of Scinc & Tchnology, Tran A, Volum, Numbr A

4 98 S Yuc / al in and co Thi prov our claim L b h volu off of For h diribuion paramr P of, w can wri From E (), i i ay o ha P d( c,, P [ P c, ], (4) whr P i h diribuion paramr of If i dvlopabl, hn h phrical indicarix vcor of i angn o i ricion curv So, from (4), w can giv h following horm wihou proof: Thorm L b h volu off of a dvlopabl ruld urfac i dvlopabl if and only if h phrical indicarix curv () of i a godic curv L b an volu off of If i clod, hn hr xi a poiiv ingr P uch ha ( P, v) (, v ) So, from (), w can giv h following horm wihou proof: Thorm L b an volu off of h clod ruld urfac wih priod P i clod if and only if R R () i a funcion wih a priod P L and b h ruld urfac which ar wp ou by h cnral normal a h corrponding ricion poin of and whr i h volu off of Thn, i i aily n ha i an volu off of Furhrmor, if w choo aympoic normal inad of h cnral normal, hn i no an volu off of g g Exampl: 4 4 ) L u conidr h clod ruld urfac (, v) (in v co, co vin, v co Th clod volu off of i (, v ) ( in v in, v co, in v in ), ( Fig ) Furhrmor, for h ruld urfac which ar wp ou by h cnral normal a corrponding ricion poin of and 4 4, w obain (,) v (in v in, co vco, v in and (, v ) ( in v co, v in, in v co ), ( Fig ) Iranian Journal of Scinc & Tchnology, Tran A, Volum, Numbr A Spring 009

5 Th involu-volu off of ruld urfac 99 Fig Clod ruld urfac and i clod volu off Fig Ruld urfac and i volu off ) L ( v, ) (co( v in(,in( v co(, v ) b a dvlopabl ruld urfac Th dvlopabl volu off of i ( v, ) ( v co( ), v in( ), 0) ( Fig ) Spring 009 Iranian Journal of Scinc & Tchnology, Tran A, Volum, Numbr A

6 00 S Yuc / al ) L i 4) Fig Dvlopabl ruld urfac and i dvlopabl volu off (,) v ( v, v, v ) 5 ( v, ) ( ( ) v, ( ) v, 4 ( ) v ) b a ruld urfac Th volu off of ( Fig Fig 4 Ruld urfac and i volu off Iranian Journal of Scinc & Tchnology, Tran A, Volum, Numbr A Spring 009

7 Th involu-volu off of ruld urfac 0 Acknowldgmn- Th auhor hank h rfr for hir carful rading of h manucrip and hlpful commn REFERENCES Alanr, R (005) Hyprruld urfac in h Minkowki 4-pac IJST, Tran A, 9(A), 4-47 Toun, M & Gungor, M A (005) A udy on im-lik complmnary ruld urfac in h Minkowki n- pac IJST, Tran A, 9(A), 5- Karadağ, H B & Klş S (005) On h ingral invarian of kinmaically gnrad ruld urfac IJST, Tran A, 9(A), Huyghn, C, (96) Horologium acillaorium, par III 5 Farouki, R T (985) Exac off procdur for impl olid Compu Aidd Gom Dign,, Farouki, R T (986) Th approximaion of non-dgnra off urfac Compu Aidd Gom Dign,, Ravani, B & Ku, TS (99) Brrand off of ruld and dvlopabl urfac Compu Aidd Gom Dign,, 45-5 Spring 009 Iranian Journal of Scinc & Tchnology, Tran A, Volum, Numbr A