Intersection of an Ellipsoid and a Plane

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1 IJRES VOLUME 6, ISSUE 6 (June, 6) (ISSN ) Internatnal Jurnal f Research n Engneerng and ppled Scences (IMPCT FCTOR 6.573) Jurnal Hme Page : Intersectn f an Ellpsd and a Plane Sebahattn Bektas Ondku Mas Unverst, Facult f Engneerng, Gematcs Engneerng, Samsun bstract The ntersectn tpc s ute ppular at an nterdscplnar level. It can be the frends f gemetr, gedes, satellte rbts n space, all srts f ellptcal mtns (e.g., planetar mtns), curvature f surfaces and cncernng ee-related rad-therap treatment, fr eample the anterr surface f the crnea s ften represented as ellpsdal n frm. We have develped an algrthm fr ntersectn f an ellpsd and a plane wth a clsed frm slutn. T d ths, we rtate the ellpsd and the plane untl nclned plane mves parallel t the plane. In ths stuatn, the ntersectn ellpse and ts prjectn wll be the same. Ths stud ams t shw hw t btan the center, the sem-as and rentatn f the ntersectn ellpse. Kewrds: ellpsd, nrmal sectn curve, 3D reverse transfrmatn, ntersectn ellpsd and a plane Intrductn The general ellpsd s treated n detal as a specal case. We wll fcus here n general ellpsd frm uadratc surfaces. Because the ellpsd s a general surface, the ellpsdal frmulas can be used easl fr rtatng ellpsd and sphere Bektas (4), Bektas (5-a). Ths ntersectn ssue s ver mprtant n gedes. T make gedetc cmputatns n the ellpsd (rtatnal r traal) frst we need t knw the nrmal sectn curve that cmbnes bservatn pnts. The nrmal sectn curve s als avalable frm the ntersectn f the ellpsd and a plane whch cntans nrmal f surface n the statn pnt and passes frm destnatn pnt. Ths current stud ams t pave the wa fr ur further stud n traal ellpsd wrk. Tda, basc nvers and frward prblem between the tw pnts n the traal ellpsd wth gedetc crdnates culd nt be a clear slutn. Our future wrk wll be n ths unslvable prblem. I hpe, the result f ths stud wll cntrbute t the slutn f the abve prblem. Internatnal Jurnal f Research n Engneerng & ppled Scences Emal:- edtrjrm@gmal.cm, (n pen access schlarl, nlne, peer-revewed, nterdscplnar, mnthl, 73

2 IJRES VOLUME 6, ISSUE 6 (June, 6) (ISSN ) Internatnal Jurnal f Research n Engneerng and ppled Scences (IMPCT FCTOR 6.573) Jurnal Hme Page : Bascall everne knws that ntersectn f a sphere and plane s a crcle. But when we get cmmn slutn the plane and the sphere euatn that wll gve us an ellpse nt a crcle. Whle the slutn n R3 space we have t elmnate ne f the, and parameters. In ths case t s clear that there wll be three pssble slutns. Three f them are generall dfferent ellpses frm each ther. Fr eample, f we elmnate the parameter, we get the fllwng ellpse euatn n the plane that s the prjectn f the true ntersectn crcle. Smlarl we fnd the ntersectn an ellpsd and a plane s an ellpse wth a cmmn slutn. But t s nt true ntersectn ellpse. That s prjectn f the true ntersectn ellpse. The ntersectng ellpse s plane s nt parallel t the plane. Ths s wh the tw ellpses are dfferent frm each ther. s related t ths subject lmted number f studes was fund n lterature. Sme f them are Klen (), Fergusn (979), URL, URL,URL3. I thnk Klen's stud s a gd stud. But understandng hs stud reures famlart wth dfferental gemetr. In ths stud we have put frward an alternatve methd addtn t the Klen's stud. We beleve that ur methd s easer than t understand Klen s. Our methd s an eas wa t understand the unfamlar dfferental gemetr. s als dfferentl, we calculate the ntersectn ellpse rentatn nfrmatn. Because the rentatn nfrmatn s etremel necessar especall n the curvature f surface. Here, ur am s t acheve the true ntersectn ellpse. T d ths, we rtate the ellpsd and the plane untl nclned plane mves parallel t the plane. In ths stuatn, the ntersectn ellpse and ts prjectn wll be the same. Of curse, n ths case we wll need t use the new euatn f ellpsd because the ellpsd s n lnger n standard pstn, t s rtated and shfted. The same stuatn s als vald fr the ntersectn f plane and rtatnal ellpsd, hperbld and ther uadratc surfaces. Defntn Ellpsd n ellpsd s a clsed uadrc surface that s analgue f an ellpse. Ellpsd has three dfferent aes (a>b>c) as shwn n Fg.. Mathematcal lterature ften uses ellpsd n place f Traal ellpsd r general ellpsd. Scentfc lterature (partcularl gedes) ften uses ellpsd n place f baal ellpsd, rtatnal ellpsd r revlutn ellpsd. Prevus lterature uses spherd n place f rtatnal ellpsd. The standard euatn f an ellpsd centered at the rgn f a Cartesan crdnate sstem and algned wth the aes as ndcated n E.. lthugh ellpsd euatn s ute smple and smth, cmputatns are ute dffcult n the ellpsd. The man reasn fr ths dffcult s the lack f smmetr. Generall, an ellpsd s defned wth 9 parameters. These parameters are; 3 crdnates f center (,,), 3 sem-aes (a,b,c) and 3 rtatnal angles (,, ) whch represent rtatns arund -,- and - aes respectvel. These angles cntrl the rentatn f the ellpsd. Internatnal Jurnal f Research n Engneerng & ppled Scences Emal:- edtrjrm@gmal.cm, (n pen access schlarl, nlne, peer-revewed, nterdscplnar, mnthl, 74

3 IJRES VOLUME 6, ISSUE 6 (June, 6) (ISSN ) Internatnal Jurnal f Research n Engneerng and ppled Scences (IMPCT FCTOR 6.573) Jurnal Hme Page : Intersectn f an Elpsd and a Plane a b c (Ellpsd euatn) () D = (Plane euatn) () Let s assume that O,O,O are the crdnates f the center f the ntersectn ellpse and a, b are the majr and mnr sem-aes f the ntersectn ellpse We can start wth the cmmn slutn f tw euatns (E.-). If we elmnate the parameter, we get the fllwng ellpse euatn n the plane that s the prjectn f the ntersectn ellpse. + B + C + D + E + F = (3) These ceffcents are calculated frm the cmmn slutn s btaned frm the tw euatns the ellpsd and plane euatn = /a + ( ) / ( c ) B= ( ) / ( c ) C= /b + ( ) / ( c ) (4) D= ( D) / ( c ) E= ( D) / ( c ) F= D / ( c ) Fg. Intersectn ellpsd and plane Internatnal Jurnal f Research n Engneerng & ppled Scences Emal:- edtrjrm@gmal.cm, (n pen access schlarl, nlne, peer-revewed, nterdscplnar, mnthl, 75

4 IJRES VOLUME 6, ISSUE 6 (June, 6) (ISSN ) Internatnal Jurnal f Research n Engneerng and ppled Scences (IMPCT FCTOR 6.573) Jurnal Hme Page : When we slve the E.3, we get fve ellpse parameters. The are: O,O (center f ellpse n plane) a, b (majr and mnr sem as f ellpse n plane) (rentatn angle between as and sem majr as ) F D / E / M D / B / E / B / C M=[ ] (5), : egenvalues f M matrces (< ) a b det( M ) /(det( M ) ) (majr sem-as f ntersectn ellpse) (6) det( M ) /(det( M ) ) (mnr sem-as f ntersectn ellpse) (7) O=(B E- C D) / (4 C-B ) (8) O=(B D- E) / (4 C-B ) (crdnates f ntersectn ellpse s center) (9) O= -( + + D ) / () (rentatn angle f prjectn ellpse) () Nw we rtate tgether the ellpsd and the plane (fg.) untl nclned plane mve parallel t the plane. In ths stuatn the ntersectn ellpse and ts prjectn wll be the same. Fr ths the rgn f the sstem must be mved t pnts f P (O,O, O). We need the transfrmatn parameters. R33-rtatn matr s btaned frm the rtatnal angles cs cs R cssn sn cssn snsn cs cs cs snsnsn - sncs snsn cssn cs sn cs cssnsn cscs () Ths s a 3D transfrmatn euatn wthut scale. Internatnal Jurnal f Research n Engneerng & ppled Scences Emal:- edtrjrm@gmal.cm, (n pen access schlarl, nlne, peer-revewed, nterdscplnar, mnthl, 76

5 IJRES VOLUME 6, ISSUE 6 (June, 6) (ISSN ) Internatnal Jurnal f Research n Engneerng and ppled Scences (IMPCT FCTOR 6.573) Jurnal Hme Page : Internatnal Jurnal f Research n Engneerng & ppled Scences Emal:- edtrjrm@gmal.cm, (n pen access schlarl, nlne, peer-revewed, nterdscplnar, mnthl, 77.. R (3) Ths transfrmatn euatn can be wrtten mre smpl wth T Epanded transfrmatn matr as fllws Bektas (5-b). T44- epanded transfrmatn matr s btaned frm the R33 rtatnal matr and the shfted parameters (O,O, O ) T = 33 R T - = R T 33 T T T (4) = T and = T - (5) Determnatn f transfrmatn parameters Shfted parameter O,O, O are ntersectn f ellpse s center crdnates that s funded befre (E.8-). We must fnd three rtatn angles (,, ). Fr ths, we take advantage f the nearest plane s pnt frm the rgn. The pnt Q n a plane D = that s clsest t the rgn has the Cartesan crdnates (,,) Shfrn and dams (). where D D D (6) nd rtatn angles (,, ) (7) arctan / (8) arctan (9) Of curse, n ths case we wll need t use the new euatn f ellpsd. Because the ellpsd s n lnger n standard pstn t s rtated and shfted. We have t reverse 3D transfrmatn t the new ellpsd parameters.

6 IJRES VOLUME 6, ISSUE 6 (June, 6) (ISSN ) Internatnal Jurnal f Research n Engneerng and ppled Scences (IMPCT FCTOR 6.573) Jurnal Hme Page : Befre we fund transfrmatn parameter O,O, O,,,. These parameters are used transfrmatn frm t. Let's see hw t fnd the reverse transfrm parameters (OT,OT, OT, T, T, T) fr the transfrmatn frm t T fnd nverse transfrmatn parameters we can take advantage f the nverse f the T matr The reverse transfrmatn parameters( OT,OT, OT, T, T, T) are lcated t T - nverse matr. Reverse shfted parameters ( OT,OT, OT) s lcated the nverse matr T - n clumn furth. Reverse rtatn angles are calculated frm the elements f the matr R as fllws T = -arc tan (R3/R33) () T = arc sn (R3) () T = -arc tan (R/R) () Nw we can wrte a new ellpsd euatn rtated and shfted, t d ths, we put (E.3). nt (E.) standard ellpsd euatn /a [(-OT) cst cs T +(-OT)(- cst) snt +(-OT) snt] + /b [(-OT)( cs T snt + sn T snt cst) + (-OT) ( cs T cst - sn T snt snt) - (-OT) sn T) cst] + /c [(-OT)( sn T snt - cs T snt cst) + (-OT)( sn T cst +cs T snt snt)+ (-OT) cs T cst] -= (3) In ths euatn f we put = we btan a cncal ntersectn ellpse euatn frm as fllws. /a [(-OT) cst cs T +(-OT)(- cst) snt -OT snt] + /b [(-OT)( cs T snt + sn T snt cst) + (-OT) ( cs T cst - sn T snt snt)+ot sn T) cst] + /c [(-OT)( sn T snt - cs T snt cst) + (-OT)( sn T cst +cs T snt snt)-ot cs T cst] -= (4) bve cnc euatn rearranged belved the ntersectn ellpse s cnc euatn btaned. + B + C + D + E + F= (5) When we slve the abve ellpse euatn, we get fve ellpse parameters f ntersectn ellpse (O, O, a, b, ). Internatnal Jurnal f Research n Engneerng & ppled Scences Emal:- edtrjrm@gmal.cm, (n pen access schlarl, nlne, peer-revewed, nterdscplnar, mnthl, 78

IJRES VOLUME 6, ISSUE 6 (June, 6) (ISSN 49-395) Internatnal Jurnal f Research n Engneerng and ppled Scences (IMPCT FCTOR 6.573) Jurnal Hme Page : http://eurasapub.rg/current.php?

Whever wants t deal wth ths subject can get n cntact wth me and use ths lnk fr free (URL3).

7 IJRES VOLUME 6, ISSUE 6 (June, 6) (ISSN ) Internatnal Jurnal f Research n Engneerng and ppled Scences (IMPCT FCTOR 6.573) Jurnal Hme Page : a)befre rtatng b) fter rtatng Fg. Rtatng tgether ellpsd and plane s a result we have presented cmputatnal results that were realed n MTLB. Whever wants t deal wth ths subject can get n cntact wth me and use ths lnk fr free (URL3).. Intersectn f a Sphere and Plane + + R = (Sphere euatn) (6) D = (Plane euatn) (7) Practcal slutns can be reached as fllws Frstl, we fnd the nearest Q pnt f plane t the rgn. The crdnates f Q pnt (,,) can be fund frm the gven plane euatn ceffcents. These crdnates were funded prevusl (E.6) The radus f the crcle ntersectns (r) r = (8) Numercal Eample- : ntersectn f ellpsd and plane (Ellpsd euatn) = (Plane euatn) Internatnal Jurnal f Research n Engneerng & ppled Scences Emal:- edtrjrm@gmal.cm, (n pen access schlarl, nlne, peer-revewed, nterdscplnar, mnthl, 79

8 IJRES VOLUME 6, ISSUE 6 (June, 6) (ISSN ) Internatnal Jurnal f Research n Engneerng and ppled Scences (IMPCT FCTOR 6.573) Jurnal Hme Page : Fnd the ntersectn f ellpsd and plane gven abve. Frstl, we fnd the euatn f cnc f prjectn ellpse n the plane. + B + C + D + E + F= These ceffcents are calculated frm the ellpsd and the plane euatn. (E.4) = B= C=.887 D= E= F= Fve ellpse parameters (,, a, b,) can be calculated frm (Es.6-) O= O= O= a = b =.7385 = The clsest pnt Q t the rgn has the Cartesan crdnates (,,), (E.6) D = D = D = nd rtatn angles (,, ) (Es.7-9). / arctan = arctan = R= Rtatn matr (E.) Internatnal Jurnal f Research n Engneerng & ppled Scences Emal:- edtrjrm@gmal.cm, (n pen access schlarl, nlne, peer-revewed, nterdscplnar, mnthl, 8

9 T= IJRES VOLUME 6, ISSUE 6 (June, 6) (ISSN ) Internatnal Jurnal f Research n Engneerng and ppled Scences (IMPCT FCTOR 6.573) Jurnal Hme Page : Epanded transfrmatn matr (E.4) T - = Inverse matr (E.4) Reverse translatn parameters ( OT,OT, OT) are lcated t the nverse matr T - n clumn 4 OT = -.37, OT =.894, OT =.69 Reverse rtatn angles are fund frm R rtatn matr (E.-) T = -arc tan (R3/R33) T = arc sn (R3) T = -arc tan ( R/R) = = 5.54 = The ntersectn ellpse n cnc frm as fllws. (E.4) = When we slve the abve ellpse euatn, we get fve ellpse parameters f ntersectn ellpse. (Es.6-) O= O= a = b = =.7799rad Numercal Eample-: ntersectn f sphere and plane + + = 5 (Sphere euatn) = (Plane euatn) Fnd the ntersectn f sphere and plane gven abve. Frstl, we fnd the nearest Q pnt f plane t the rgn. The crdnates f Q pnt (,,) can be fund frm the gven plane euatn ceffcents as belw. Internatnal Jurnal f Research n Engneerng & ppled Scences Emal:- edtrjrm@gmal.cm, (n pen access schlarl, nlne, peer-revewed, nterdscplnar, mnthl, 8

10 IJRES VOLUME 6, ISSUE 6 (June, 6) (ISSN ) Internatnal Jurnal f Research n Engneerng and ppled Scences (IMPCT FCTOR 6.573) Jurnal Hme Page : D = D = D = The radus f the crcle ntersectns (r) r = = Cnclusn In ths stud, we have develped an algrthm fr ntersectn f an ellpsd and a plane wth a clsed frm slutn. The effcenc f the new appraches s demnstrated thrugh a numercal eample. The presented algrthm can be appled easl fr spherd, sphere and als ther uadratc surface, such as parabld and hperbld. References [ ] Bektas, S, (4) Orthgnal Dstance Frm n Ellpsd, Bletm de Cencas Gedescas, Vl., N. 4 ISSN 98-7, 4 Bektas, S, (5a), Least suares fttng f ellpsd usng rthgnal dstances, Bletm de Cencas Gedescas, Vl., N. ISSN 98-7, 759, Bektas, S, (5b), Gedetc Cmputatns n Traal Ellpsd, Internatnal Jurnal f Mnng Scence (IJMS) Vlume, Issue, June 5, PP RC Page 5 C. C. Fergusn, (979),Intersectns f Ellpsds and Planes f rbtrar Orentatn and Pstn, Mathematcal Gelg, Vl., N. 3, pp d:.7/bf34997 Klen,P. (), On the Ellpsd and Plane Intersectn Euatn, ppled Mathematcs, Vl. 3 N., pp d:.436/am..36 Shfrn, Ted; dams, Malclm(), Lnear lgebra: Gemetrc pprach ( ( nd ed.), Macmllan, p. 3, ISBN URL The Math Frum, Ellpsd and Plane Intersectn Euatn,. ccessed 6 March 6 Internatnal Jurnal f Research n Engneerng & ppled Scences Emal:- edtrjrm@gmal.cm, (n pen access schlarl, nlne, peer-revewed, nterdscplnar, mnthl, 8

11 IJRES VOLUME 6, ISSUE 6 (June, 6) (ISSN ) Internatnal Jurnal f Research n Engneerng and ppled Scences (IMPCT FCTOR 6.573) Jurnal Hme Page : URL The Math Frum, Intersectn f Hperplane and an Ellpsd, 7. ccessed 6 March 6 URL3 ccessed 6 March 6. Intersectn f an Ellpsd and a Plane Sebahattn Bektas Internatnal Jurnal f Research n Engneerng & ppled Scences Emal:- edtrjrm@gmal.cm, (n pen access schlarl, nlne, peer-revewed, nterdscplnar, mnthl, 83

On the Intersection of a Hyperboloid and a Plane

On the Intersection of a Hyperboloid and a Plane Internatinal Jurnal f Discrete Mathematics 2017; 2(2): 38-42 http://www.sciencepublishinggrup.cm/j/dmath di: 10.11648/j.dmath.20170202.12 On the Intersectin f a Hperblid and a Plane Sebahattin Bektas Facult