ANALYTICAL AND GRAPHICAL SOLUTIONS TO PROBLEMS IN DESCRIPTIVE GEOMETRY INVOLVING PLANES AND LINES

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1 ULETINUL INSTITUTULUI POLITENI DIN IŞI Publicat de Uniersitatea Tenică George saci din Iaşi Tomul LVII (LXI) Fasc 3 0 Secţia ONSTRUŢII DE MŞINI NLYTIL ND GRPIL SOLUTIONS TO PROLEMS IN DESRIPTIVE GEOMETRY INVOLVING PLNES ND LINES Y PUL GEORGESU bstract Seeral common situations wic occur wen soling problems in descriptie geometr including finding te intersection of two planes passing troug points wit coordinates of a certain form and finding te intersections between particular lines and planes are inestigated troug analtical and grapical metods e words: planes lines traces analtic geometr descriptie geometr Te statement of te problem Of concern in tis paper is te following problem encompassing seeral situations commonl occuring in concrete applications of descriptie geometr Gien te points (00030) (60500) (60040) M(0080) N(60800) R (000) I(80) find: a) te coordinates of I knowing tat tis point is situated on te () line; b) te traces of te plane [P] defined b te lines (I) and (); c) te intersection between te planes [P] and [Q] were [Q] is a plane parallel to (O) wic as traces at distance 5 and eleation 65 respectiel; d) te intersection point between te line ( ) and te plane [ Q ] We sall sole tis problem first b means of analtic geometr briefl discussing te main teoretical points and te formulae concerned and ten b means of descriptie geometr

2 Paul Georgescu Te analtical approac We first discuss seeral properties of a plane [ Q ] wic is parallel to ( O ) and passes troug Q (0 d0) of distance d and Q (00 e) of eleation e respectiel Namel we sall find te equation of [ Q ] and determine its intersection wit gien lines and planes Te equation of [Q] plane Since two ectors parallel to [ Q ] are i and QQ = dj+ ek te ector nq = i QQ = dk ej is normal to [ Q ] Te equation of [ Q ] is ten d e + D = 0 te constant D being determined troug te condition tat Q belongs to [ Q ] onsequentl te equation of [ Q ] can be written as: () [ Q]: e+ d de= 0 Te intersections of [ Q ] wit lines and planes We find now te intersection of a line determined b two of its points ( ) ( ) and te plane [ Q ] One sees tat te parametric equation of te line ( ) is = = = t tat is [] [5]: () ) : = + t( ) = + t( ) = + t( ) ( t being a real parameter To find te intersection between ( ) and [ Q ] we sole te sstem consisting in Eqs () and () substituting and gien b Eq () into Eq () we find tat: (3) de e d t = e d ( ) + ( ) onsequentl te point of intersection between ( ) and [ Q ] as coordinates: (4) de e d = + e ( ) ( ) ( ) + d

3 ul Inst Polit Iaşi t LVII (LXI) f (5) (6) de e d = + e ( ) ( ) ( ) + d de e d = + e ( ) ( ) ( ) + d We find now te intersection between te [ Q ] plane and a [ P] plane determined b ( ) ( ) and ( ) Tis line can be tougt as being determined b te point aboe te intersection between ( ) and [ Q ] and te point te intersection between () and [ Q ] similarit wit Eqs (4)-(6) it is seen tat te coordinates of are gien b: () (8) (9) de e d = + e ( ) ( ) ( ) + d de e d = ( ) ( ) ( ) + e + d de e d = + e + d ( ) ( ) ( ) Ten te intersection ( ) ( D) between [ P] and [ Q ] as te parametric equation: (0) = = = t t R and being gien b Eqs (4)-(6) and ()-(9) 3 Te traces of a plane [P] In wat follows we sall determine te traces of a plane determined b tree of its points on te coordinate planes It can be sown [] tat te traces of a plane [ P ] determined b ( ) ( ) and ( ) ae te following equations: () ( P ) : Δ + Δ = Δ; = 0 () ( P): Δ + Δ =Δ ; = 0 l (3) ( P ): Δ + Δ =Δ ; = 0

4 4 Paul Georgescu lso [P] intersects te coordinate aes in te points P P P gien b: (4) Δ P = P 00 Δ Δ P Δ = P 0 0 P = P Δ 0 0 Δ (5) Δ = Δ = Δ = (6) Δ = We are now read to sole te problem stated in Section S o l u t i o n a) Since I is situated on te line ( ) its coordinates I I I erif Eq () for a alue of t wic is to be determined From te first part of Eq () it follows tat t = and consequentl tat I = 30 and I = 0 b) Te plane [ P ] can be tougt as being determined b and onsequentl since Δ = Δ = 600 Δ = 00 and Δ = 00 it follows ten from Eq (4) tat te intersections of [ P ] wit te coordinate aes are P ( 8000) P ( 0900) P ( 0090) O is ( P ) : 90; 0 It also follows from Eqs ()-(3) tat te trace of te plane [ P ] on ( ) + = = te trace of [ P ] on ( O ) is ( P ):+ = 90; = 0 and te trace of [ P ] on ( O ) is ( P): + = 90; = 0 l 5 c) Using Eqs (4)-(6) and Eqs ()-(9) it follows tat = 90 = = and = = = From Eq (0) one ma deduce tat te intersection between te plane [ P ] and te plane [ Q ] is te line ( ) wic as te parametric equation:

5 ul Inst Polit Iaşi t LVII (LXI) f () 60 6 = = 3 = t t R 5 For = 0 it follows tat t = and consequentl te intersection between ( ) and te plane (O) is te point (3050 ) Similarl for = 0 it 0 follows tat t = and te intersection between ( ) and te plane (O) is te point V (50065 ) d) s seen aboe = = = 3 Descriptie Geometr approac a) ecause te point I is ling on te straigt line () its projections are located on te corresponding projections of te straigt line (Fig ) [4] ence if a point I () it follows tat i (ab) i ( a b ) and i ( a b ) Z(-Y) X a' a i' b' a'' i'' b'' Y(-X) i b Y(-Z) Fig Projections of te point I [3]

6 l 6 Paul Georgescu Z(-Y) ' P X P ' a' ' a i' c c' b' ' l Y(-X) P i b Y(-Z) P Fig Traces of te plane [P] b) We determine first te projections and b etending te projections (ab) (ci) and (a'b') (c'i') respectiel up to te (O) ais and ten b using te projecting lines we find te traces of gien straigt lines: on (ab) on (ci) and ' on (a'b') on (c'i') respectiel (Fig ) Z(-Y) P ' l X P ' (d) (d') (d") Q Y(-X) P Y(-Z) Fig 3 (D)(dd'd'') te intersection line of planes [P] and [Q] P

7 l ul Inst Polit Iaşi t LVII (LXI) f 3 0 c) Two concurrent planes [P] and [Q] intersect along a straigt line (D)(dd'd'') (Fig 3) wose traces lie at te intersection of corresponding traces of bot planes (R ) ' 3 (δ ) Q Z(-Y) X R a' 3 a k' a'' k" k ' 3 b' (δ ) b'' Y(-X) 3 Q b (R ) (δ) Y(-Z) Fig 4 te intersection point of ()-[Q] intersection d) We draw an auiliar cutting plane [R] wic contains te gien straigt line and we find te line of intersection of te cutting plane [R] and te gien plane [Q] (Fig 4) Te required point ( k k k ) is at te intersection of te straigt lines ( )( ab a b a b ) and ( Δ )( δ δ δ ) common to bot planes [P] and [R] 4 onclusions s we can see from a comparatie iew of bot solutions presented aboe te one emploing Descriptie Geometr offers a quick and elegant reasoning wic appeals to one s intuition wile te approac based on naltic Geometr altoug more computational offers general formulas wic can be applied to a large class of problems wic are reductible to te ones discussed in tis paper 00 George saci Tecnical Uniersit Department of Matematics Iaşi Romania pgeorgescu@gmailcom

8 8 Paul Georgescu R E F E R E N E S ă r ă u ş u Vector algebra analtic & differential geometr Ed PIM Iaşi 003 G e o r g e s c u P î n c u G Finding te traces of a gien plane: analticall and troug grapical constructions Proceedings of IEGD0 Sustainable Eco-Design -4 june 0 Iaşi ul Inst Polit Iaşi LVII (LXI) Facultatea de onstrucţii de Maşini (0) 3 î n c u G Infografică utod Ed Societăţii cademice Matei Teiu ote Iaşi î n c u G Geometrie descriptiă urs şi aplicaţii Ed PIM Iaşi * * * Mica Enciclopedie Matematică Ed Tenică ucureşti 985 SOLUŢII NLITIE ŞI GRFIE LE UNOR PROLEME DIN GEOMETRI DESRIPTIVĂ ÎN RE INTERVIN PLNE ŞI DREPTE (Reumat) Lucrarea de faţă abordeaă câtea probleme de intersecţie ale unor drepte şi plane particulare prin metodele geometriei analitice şi respecti ale geometriei descriptie Un studiu comparati al celor două metode atestă faptul că raţionamentul baat pe geometrie descriptiă este mai intuiti mai rapid în reme ce raţionamentul analitic deşi mai laborios poate fi adaptat pentru reolarea unei clase largi de probleme înrudite