On the Intersection of a Hyperboloid and a Plane

Transcription

1 Internatinal Jurnal f Discrete Mathematics 2017; 2(2): di: /j.dmath On the Intersectin f a Hperblid and a Plane Sebahattin Bektas Facult f Engineering, Gematics Engineering, Ondkuz Mais Universit, Samsun, Turke address: sbektas@mu.edu.tr T cite this article: Sebahattin Bektas. On the Intersectin f a Hperblid and a Plane. Internatinal Jurnal f Discrete Mathematics. Vl. 2, N. 2, 2017, pp di: /j.dmath Received: Januar 19, 2017; Accepted: Februar 7, 2017; Published: March 1, 2017 Abstract: The intersectin tpic is uite ppular at an interdisciplinar level. It can be the friends f gemetr, gedes and thers. The curves f intersectin resulting in this case are nt nl ellipses but rather all tpes f cnics: ellipses, hperblas and parablas. In text bks f mathematics usuall nl cases are treated, where the planes f intersectin are parallel t the crdinate planes. Here the general case is illustrated with intersecting planes which are nt necessaril parallel t the crdinate planes. We have develped an algrithm fr intersectin f a hperblid and a plane with a clsed frm slutin. T d this, we rtate the hperblid and the plane until inclined plane mves parallel t the X plane. In this situatin, the intersectin ellipse and its prjectin will be the same. This stud aims t shw hw t btain the center, the semi-axis and rientatin f the intersectin curve. Kewrds: Hperblid, Intersectin, 3D Reverse Transfrmatin, Plane 1. Intrductin 1.1. Definitin f Hperblid Let a hperblid be given with the three psitive semi axes a, b, c see Fig. 1 X Z + ± 1 (Hperblid euatin) (1) a b c hperblid f ne sheet, n the right hand side f -1 t a hperblid f tw sheets Parameterizatin f Hperblid Cartesian crdinates fr the hperblids can be defined, similar t spherical crdinates, keeping the azimuth angle θ [0, 2π), but changing inclinatin v int hperblic trignmetric functins: One-surface hperblid: v (, ) +1 where n the right hand side f (1) crrespnds t a a)hperblid f ne sheet Figure 1. Hperblid. b)hperblid f tw sheets

2 Internatinal Jurnal f Discrete Mathematics 2017; 2(2): Generalised Euatin f Hperblid Mre generall, an arbitraril riented hperblid, centered at v, is defined b the euatin (x-v) T A (x-v) 1 where A is a matrix and x, v are vectrs. The eigenvectrs f A define the principal directins f the hperblid and the eigenvalues f A are the reciprcals f the suares f the semi-axes: 1/a 2, 1/b 2 and 1/c 2. The ne-sheet hperblid has tw psitive eigenvalues and ne negative eigenvalue. The tw-sheet hperblid has ne psitive eigenvalue and tw negative eigenvalues. [5], [13], [4] Basicall everne knws that intersectin f a sphere and plane is a circle. But when we make a cmmn slutin f the sphere euatin with the plane euatin, it will give us an ellipse that is nt a circle. While the slutin in R3 space we have t eliminate ne f the X, and Z parameters. In this case it is clear that there will be three pssible slutins. Three f them are generall different ellipses frm each ther. Fr example, if we eliminate the parameter Z, we get the fllwing ellipse euatin n the X plane that is the prjectin f the true intersectin circle. Similarl we find the intersectin a Hperblid and a plane is an ellipse with a cmmn slutin. But it is nt true curve. That is prjectin f the true intersectin curve. The intersecting curve s plane is nt parallel t the X plane. This is wh the tw curves are different frm each ther. In this stud, we will be shwn hw t btain the center, the semi-axis and rientatin f the intersectin curve. As related t this subject limited number f studies was fund in literature. Sme f them are [9], [6] [10], [16], [15]. I think Klein's stud [9] is a gd stud. But understanding his stud reuires familiarit with differential gemetr. In this stud we have put frward an alternative methd additin t the Klein's stud. We believe that ur methd is easier than t understand Klein s. Our methd is an eas wa t understand the unfamiliar differential gemetr. As als differentl, we calculate the intersectin curve s rientatin infrmatin. Because the rientatin infrmatin is extremel necessar especiall in the curvature f surface. Here, ur aim is t achieve the true intersectin curve. T d this, we rtate the Hperblid and the plane until inclined plane mves parallel t the X plane. In this situatin, the intersectin curve and its prjectin will be the same. Of curse, in this case we will need t use the new euatin f Hperblid because the Hperblid is n lnger in standard psitin, it is rtated and shifted. The same situatin is als valid fr the intersectin f plane and rtatinal ellipsid and ther uadratic surfaces. Generall, a Hperblid is defined with 9 parameters. These parameters are; 3 crdinates f center (x,,z ), 3 semi-axes (a,b,c) and 3 rtatinal angles (ε, ψ, ω) which represent rtatins arund x-,- and z- axes respectivel. These angles cntrl the rientatin f the hperblid. 2. Intersectin f an Hperblid and a Plane The intersectin tpic is uite ppular at an interdisciplinar level. It can be the friends f gemetr, gedes and thers. The curves f intersectin resulting in this case are nt nl ellipses but rather all tpes f cnics: ellipses, hperblas and parablas. In text bks f mathematics usuall nl cases are treated, where the planes f intersectin are parallel t the crdinate planes. Here the general case is illustrated with intersecting planes which are nt necessaril parallel t the crdinate planes. We have develped an algrithm fr intersectin f a hperblid and a plane with a clsed frm slutin. T d this, we rtate the hperblid and the plane until inclined plane mves parallel t the X plane. In this situatin, the intersectin ellipse and its prjectin will be the same. This stud aims t shw hw t btain the center, the semi-axis and rientatin f the intersectin curve. Here plane euatin A X X + A + A Z Z + A D 0 (Plane euatin) (2) 2.1. The Classificatin f Intersectin It appears that the classificatin f hperblic cnic sectins ges back t an 1882 paper b William E. Str [13]: Str classifies cnic sectins accrding t the number and multiplicities f intersectins between the cnic sectin and the bundar circle. This results in eight tpes f cnic sectins 1. An ellipse is an ellipse cntained entirel within the interir f the unit disk. 2. A hperbla is a cnic sectin that intersects the unit circle at fur different pints. (Such a cnic sectin ma be a prtin f an ellipse, parabla, r hperbla in the Euclidean plane, thugh which f these three tpes it is ma change under hperblic ismetries.) 3. A semi-hperbla is a cnic sectin that intersects the unit circle transversel at tw different pints. (Again, this ma be an ellipse, parabla, r hperbla in the Euclidean plane.) 4. An elliptic parabla is an ellipse r circle in the disk that intersects the unit circle at ne pint f tangenc. 5. A hperblic parabla is a cnic sectin that intersects the unit circle three times, with ne being a pint f tangenc. 6. A semi-circular parabla is a cnic sectin that has the unit circle as ne f its sculating circles (i.e. the have a pint f third-rder cntact) and als intersects the unit circle at ne additinal pint. 7. A hrccle (called a circular parabla b Str) is an ellipse in the unit disk that has a furth-rder cntact with the unit circle. That is, it is an ellipse that has the unit circle as an sculating circle at ne f the endpints f its minr axis.

3 40 Sebahattin Bektas: On the Intersectin f a Hperblid and a Plane 8. A circle is a circle cntained entirel in the interir f the unit disk, and an euidistance cnic is an ellipse that is tangent t the unit disk at tw pints. (Str refers t bth f these cases simpl as circles.) Str mentins that sme f these eight tpes can be further subdivided, and later authrs ften increased the number f tpes t 11 r 12. Fr example, a classificatin int 12 tpes can be fund n pg. 142 in [14]. Let s assume that X O, O, Z O are the crdinates f the center f the intersectin curve (ellipses, hperblas and parablas) and a i, b i are the majr and minr semi-axes f the intersectin curve. We can start with the cmmn slutin f tw euatins (E.1-2). If we eliminate the parameter Z, we get the fllwing ellipse euatin n the X plane that is the prjectin f the intersectin curve. A X 2 + B X + C 2 + D X + E + F 0 (3) These cefficients are calculated frm the cmmn slutin is btained frm the tw euatins the hperblid and plane euatin. A 1/a 2 + (A x 2 ) / (A z 2 c 2 ) B (2 A x A ) / (A z 2 c 2 ) C 1/b 2 + (A 2 ) / (A z 2 c 2 ) (4) D (2 A x A D ) / (A z 2 c 2 ) E (2 A A D ) / (A z 2 c 2 ) F A D 2 / (A z 2 c 2 ) 1 When we slve the E.3, we get five ellipse parameters. The are: X O, O (center f ellipse in X plane) a, b (majr and minr semi axis f ellipse in X plane) θ (rientatin angle between X axis and semi majr axis) F D / 2 E / 2 M D / 2 A B / 2 E / 2 B / 2 C λ 1, λ 2 : eigenvalues f M matrices (λ 1 < λ 2 ) a det( M ) / (det( M) λ ) M /2 /2 (5) (majr semi-axis f intersectin ellipse) (6) b det( M ) / (det( M) λ ) (minr semi-axis f intersectin ellipse) (7) X O (B E-2 C D) / (4 A C-B 2 ) (8) O (B D-2 A E) / (4 A C-B 2 ) (9) (crdinates f intersectin ellipse s center) Z O -(A x X + A + A D ) / A z (10) 1 2 tan2θ (11) (rientatin angle f prjectin ellipse) Nw we rtate tgether the hperblid and the plane until inclined plane mve parallel t the X plane. In this situatin the intersectin ellipse and its prjectin will be the same. Fr this the rigin f the XZ sstem must be mved t pints f P (X O, O, Z O ). We need the transfrmatin parameters. R 3x3 -rtatin matrix is btained frm the rtatinal angles csψ cs ω csεsinω + sinεsinψ cs ω sinεsinω csεsinψ csω R csψ sin ω csε csω sinεsinψ sin ω sinε csω + csεsinψ sin ω sin ψ -sinεcs ψ csεcs ψ (12) This is a 3D transfrmatin euatin withut scale. X X x. R. + Z z Z (13) This transfrmatin euatin can be written mre simpl with T Expanded transfrmatin matrix as fllws [1, 2, 3]. T 4x4 - expanded transfrmatin matrix is btained frm the R 3x3 rtatinal matrix and the shifted parameters (X O, O, Z O ) T X 0 R 3x3 0 Z T -1 X 0T T R3x3 0T Z0T Xi xi xi i i T and i T -1 Zi zi zi Xi i Zi Determinatin f Transfrmatin Parameters (14) (15) Shifted parameter X O, O, Z O are intersectin f ellipse s center crdinates that is funded befre (E.8-10). We must find three rtatin angles (ε, Ψ, ω). Fr this, we take advantage f the nearest plane s pint frm the rigin. The pint Q n a plane A X.X + A. + A Z.Z + A D 0 that is

4 Internatinal Jurnal f Discrete Mathematics 2017; 2(2): clsest t the rigin has the Cartesian crdinates ( x,, z ) [11]. Where x x D x + + Az And rtatin angles (ε, Ψ, ω) z ψ π / 2 arctan z D x + + z A D x + + z A (16) ε 0 (17) z 2 2 x + ω π arctan x (18) (19) Of curse, in this case we will need t use the new euatin f hperblid. Because the hperblid is n lnger in standard psitin it is rtated and shifted. We have t reverse 3D transfrmatin t the new hperblid parameters. Befre we fund transfrmatin parameter X O, O, Z O, ε, Ψ, ω. These parameters are used transfrmatin frm XZ t xz. Let's see hw t find the reverse transfrm parameters (X OT, OT, Z OT, ε T, Ψ T, ω T ) fr the transfrmatin frm xz t XZ T find inverse transfrmatin parameters we can take advantage f the inverse f the T matrix The reverse transfrmatin parameters (X OT, OT, Z OT, ε T, Ψ T, ω T ) are lcated t T -1 inverse matrix. Reverse shifted parameters (X OT, OT, Z OT ) is lcated the inverse matrix T -1 in clumn furth. Reverse rtatin angles are calculated frm the elements f the matrix R as fllws ε T -arc tan (R 23 /R 33 ) (20) ψ T arc sin (R 13 ) (21) ω T -arc tan (R 12 /R 11 ) (22) Nw we can write a new hperblid euatin rtated and shifted, t d this, we put (E.13). int (E.1) standard hperblid euatin 1/a 2 [(X-X OT ) csψ T csω T +(- OT )(- csψ T ) sinω T +(Z- Z OT ) sinψ T ] 2 + 1/b 2 [(X-X OT )(csε T sinω T + sinε T sinψ T csω T ) + (- OT ) (csε T csω T - sinε T sinψ T sinω T ) - (Z-Z OT ) sinε T ) csψ T ] 2-1/c 2 [(X-X OT )( sinε T sinω T - csε T sinψ T csω T ) + (- OT )( sinε T csω T +csε T sinψ T sinω T )+ (Z-Z OT ) csε T csψ T ] 2 ±1 (23) In this euatin if we put z 0 we btain a cnical intersectin ellipse euatin frm as fllws. 1/a 2 [(X-X OT ) csψ T csω T +(- OT )(- csψ T ) sinω T -Z OT sinψ T ] 2 + 1/b 2 [(X-X OT )( csε T sinω T + sinε T sinψ T csω T ) + (- OT ) ( csε T csω T - sinε T sinψ T sinω T )+Z OT sinε T ) csψ T ] 2-1/c 2 [(X-X OT )( sinε T sinω T - csε T sinψ T csω T ) + (- OT )( sinε T csω T +csε T sinψ T sinω T )-Z OT csε T csψ T ] 2 ±1 (24) Abve cnic euatin rearranged belved the intersectin ellipse s cnic euatin btained. A X 2 + B X + C 2 + D X + E + F0 (25) When we slve the abve ellipse euatin, we get five ellipse parameters f intersectin ellipse (X O, O, a i, b i, θ). As a result we have presented cmputatinal results that were realized in MATLAB. 3. Cnclusin In this stud, we have develped an algrithm fr intersectin f a hperblid and a plane with a clsed frm slutin. The efficienc f the new appraches is demnstrated thrugh a numerical example. The presented algrithm can be applied easil fr spherid, sphere and als ther uadratic surface, such as parablid and ellipsid. References [1] Bektas, S, (2014) Orthgnal Distance Frm An Ellipsid, Bletim de Ciencias Gedesicas, Vl. 20, N. 4 ISSN [2] Bektas, S, (2015a), Least suares fitting f ellipsid using rthgnal distances, Bletim de Ciencias Gedesicas, Vl. 21, N. 2 ISSN , [3] Bektas, S, (2015b), Gedetic Cmputatins n Triaxial Ellipsid, Internatinal Jurnal f Mining Science (IJMS) Vlume 1, Issue 1, June 2015, PP ARC Page 25. [4] Cxeter, H. S. M. (1961) Intrductin t Gemetr, p. 130, Jhn Wile & Sns. [5] David A. Brannan, M. F. Esplen, & Jerem J Gra (1999) Gemetr, pp Cambridge Universit Press. [6] C. C. Fergusn, (1979), Intersectins f Ellipsids and Planes f Arbitrar Orientatin and Psitin, Mathematical Gelg, Vl. 11, N. 3, pp di: /bf [7] Hawkins T. (2000) Emergence f the Ther f Lie Grups: an essa in the histr f mathematics, , 9.3 "The Mathematizatin f Phsics at Göttingen", see page 340, Springer ISBN [8] Heinrich, L (1905). Nichteuklidische gemetrie. Vl. 49. GJ Göschen, 1905.

5 42 Sebahattin Bektas: On the Intersectin f a Hperblid and a Plane [9] Klein, P. (2012), On the Hperblid and Plane Intersectin Euatin, Applied Mathematics, Vl. 3 N. 11, pp di: /am [10] Prteus, R (1995) Cliffrd Algebras and the Classical Grups, pages 22, 24 & 106, Cambridge Universit Press ISBN [11] Shifrin, Ted; Adams, Malclm (2010), Linear Algebra: A Gemetric Apprach ( 32) (2 nd ed.), Macmillan, p. 32, ISBN [12] Str, William E. (1882) On nn-euclidean prperties f cnics. American Jurnal f Mathematics 5, n. 1 (1882): [13] Wilhelm, B. (1948) Analtische Gemetrie, Kapital V: "Quadriken", Wlfenbutteler Verlagsanstalt. [14] H. Liebmann, Nichteuklidische Gemetrie, B. G. Teubner, Leipzig, [15] URL 1 The Math Frum, Ellipsid and Plane Intersectin Euatin, Accessed 26 March [16] URL 2 The Math Frum, Intersectin f Hperplane and an Ellipsid, Accessed 26 March 2016.