BULETINUL INSTITUTULUI POLITEHNI DIN IŞI Publicat de Universitatea Tehnică Gheorghe sachi din Iaşi Tomul LVII (LXI), Fasc. 3, 20 Secţia ONSTRUŢII DE MŞINI FINDING THE TRES OF GIVEN PLNE: NLYTILLY ND THROUGH GRPHIL ONSTRUTIONS BY PUL GEORGESU and GRŢIEL HÎNU 2 bstract. This paper determines eplicitel the traces of a plane [P] given b three of its points, using two different approaches: b means of analtic geometr and descriptive geometr methods, respectivel. Ke words: traces, planes, analtic geometr, descriptive geometr.. Introduction The problem which is of concern in this paper is stated as follows: Find the traces of the plane [P] determined b the point (70, 50, 40) and the straight line(b), where B(00, 20, 50) and (0, 80, 0). In what follows, this problem will be solved first b using analtic geometr and then b methods of descriptive geometr, a comparison between these approaches being drawn as a conclusion. 2. Solving Methods 2.. naltical Geometr Method Having in view that a line is completel determined b two of its points, we solve the general problem of determining the traces of a plane given b three points (,, ), B ( B, B, B), (,, ). Then, we particularie the results for the concrete problem stated above. We start b determining the trace ( P h ) of the plane [P] on the plane ( O ). First, let us denote:
2 Paul Georgescu () Δ =, Δ =, Δ =, B B B B B B (2) Δ=. B B B Since the equation of [P] is: (3) B B B = 0 and all points in the plane ( O ) have the -coordinate equal to 0, one obtains b epanding the determinant along the first row that: (4) ( P ): = 0; = 0. h B B B B B B B that is, (5) ( P ): + = ; = 0. h B B B B B B B onsequentl, the trace ( P h ) of [P] on the plane ( O ) is characteried b: () ( P ): Δ + Δ =Δ ; = 0. h
Bul. Inst. Polit. Iaşi, t. LVII (LXI), f. 3, 20 3 Through the circular permutation, one obtains that the trace ( P l ) of [P] on the plane ( O ) is given b: (7) ( P): Δ + Δ =Δ ; = 0 l and the trace ( P v ) of [P] on the plane ( O ) is given b: (8) ( P ): Δ + Δ =Δ ; = 0. v Note that the traces ( P h ), ( P l ) and ( P v ) can also be given in the following parametric forms: Δ Δ Δ Δ (9) ( Ph ) : = t; = + t; = 0, t R, 2 Δ Δ 2 Δ Δ Δ Δ Δ Δ (0) ( Pv ) : = t; = + t; = 0, t R, 2 Δ Δ 2 Δ Δ Δ Δ Δ Δ () ( Pl ) : = t; = + t; = 0, t R. 2 Δ Δ 2 Δ Δ Then, we determine the intersections between [P] and the coordinate aes. Since the intersection between [P] and (O) is actuall the intersection between its trace ( P h ) and (O) and this point has the -coordinate equal to 0, one obtains from Eq () that the intersection between [P] and (O) is the point P given b: (2) P = P Δ,0,0. Δ Through circular permutations, one obtains that the intersection between [P] and (O) is the point P given b: (3) P P Δ = 0,,0 Δ
4 Paul Georgescu and the intersection between [P] and (O) is the point P given b: (4) P = P 0,0, Δ. Δ We are now read to solve the problem stated in the Introduction. S o l u t i o n One sees that: (5) () (7) (8) 70 50 40 Δ= = 00 20 50 = 0000, B B B 0 80 0 70 50 Δ = = 00 20 = 00, B B 0 80 50 40 Δ = = 20 50 = 00, B B 80 0 40 70 Δ = = 50 00 = 800. B B 0 0 From Eqs (2)-(4), it follows that the intersections of [P] with the coordinate 25 aes are: P,0,0, P 0,,0 and P 0,0,. From Eqs (5)-(7), it 3 2 3 also follows that the trace ( P h ) is 3 + 4 = ; = 0, the trace ( P v ) is 3 + 3 = ; = 0 and the trace ( P l ) is 4 + 3 = ; = 0. lternativel, form Eqs (9)-, the parametric equations of the traces are: 25 (8) ( P h ) : = ( 2t); = ( + 2t); = 0, t R, 4
Bul. Inst. Polit. Iaşi, t. LVII (LXI), f. 3, 20 5 (9) 25 ( P v ) : = ( 2t); = ( + 2t); 4 = 0, t R, (20) ( P l ) : = ( 2t); = ( + 2t); = 0, t R. 2.2. Descriptive Geometr Method Z(-Y) P (P v) v' v" X P v' v b' b v (d) a' c' (d') h' h' v" b'' (d'') a'' (P ) l c'' h" h" P Y(-X) (P ) h a c h h Y(-Z) P Fig. Traces of [P] plane We determine the traces h and v of the straight line ( B ) and then we construct through the point a line (D ) which is parallel to the given one (Fig. ). To determine the traces of the plane, we need onl one of the traces of the line (D ) (for eample, the horiontal trace h ). First of all, we represent the horiontal trace ( P h ) of [P] and then the vertical one, ( P v ). We note that P =P =7.() and P =32.50, which are the values obtained b using the analtical method.
Paul Georgescu 3. onclusions s we can see, Descriptive Geometr offers a quick and elegant solution to the problem, while the analtical approach, although more computational, offers general formulas which can be applied to a large class of related problems. 22.0.20 Gheorghe sachi Technical Universit, Department of Mathematics Iaşi, Romania, e-mail: v.p.georgescu@gmail.com 2 Gheorghe sachi Technical Universit Department of Descriptive Geometr and Technical Drawing Iaşi, Romania, e-mail: hgratiela@ahoo.com R E F E R E N E S. ă r ă u ş u., Vector algebra, analtic & differential geometr. Ed. PIM, Iaşi, 2003. 2. * * * Mica Enciclopedie Matematică, Editura Tehnică, Bucureşti, 985. DETERMINRE URMELOR UNUI PLN FOLOSIND METODELE GEOMETRIEI NLITIE ŞI LE GEOMETRIEI DESRIPTIVE (Reumat) Lucrarea îşi propune să preinte modalităţi diferite de determinare a urmelor unui plan dat prin trei puncte necoliniare, abordând metodele geometriei analitice şi aplicând principiile geometriei descriptive. Deşi ambele metode ajung la acelaşi reultat privind determinarea urmelor planului [P], geometria descriptivă este mai rapidă şi mai intuitivă decât geometria analitică, care necesită calcule laborioase, dar are o aplicabilitate etinsă. Sperăm că demersul nostru va fi util, interesant şi edificator pentru studenţii facultăţilor tehnice, dar şi celor interesaţi de cele două discipline de studiu.