DEFINING CORRESPONDING IDENTICAL TUFTS IN COLOCAL GENERAL COLINEAR FIELDS UDC (045) Sonja Krasić

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1 FACTA UNIVERSITATIS Series: Architecture and Civi Engineering Vo. 2, N o 1, 1999, pp DEFINING CORRESPONDING IDENTICAL TUFTS IN COLOCAL GENERAL COLINEAR FIELDS UDC (045) Sonja Krasić Facuty of Civi Engineering and Architecture, Beogradska 14, 18000,YU Niš E-mai: sonjak@mai.gaf.ni.ac.yu Abstract. Cooca coinear fieds can be brought from genera to a perspective position, where the procedure of copying can be simpified. If, among projected tufts, there are identica ones that are at the same time the corresponding tufts, then their coinciding brings the fieds into the perspective position. The procedure of defining corresponding identica tufts is in finding the tufts that are identica to a other tufts in the set of ² perspective tufts in the second fied (whose axis is the endessy distant ine), in the set of ² perspective tufts in the first fied (whose axis of the perspective is the infinite ine). The starting point is a genera situation, I. e. the rays are arbitrariy taken in the first fied. Then the procedure gets simpified by introducing the speciay taken rays in the tufts in the first fied. The corresponding identica tufts beong to invariants of genera coinear and perspective coinear fieds because of their characteristics, which ony depend on projective capacity given by the four of homogenous corresponding points (ines). The concusion is that the two corresponding pairs of identica tufts do exist in the genera coinear fieds. Key words: Cooca genera coinear fieds, corresponding identica tufts. INTRODUCTION Generay, Cooca coinear fieds are given with four pairs of corresponding points, i. e. ines, by which the projecting reation is introduced. Copying from one fied into another demands quite a compex graphic presentation. The copying procedure is simpified in the cooca coinear fieds in the perspective position. How can cooca coinear fieds be brought from a genera to a perspective position? If corresponding identica tufts exist, then their coincidence produces a doube tuft and the coinciding tops of the tufts (the doube tuft carrier) become the perspective centre in the perspective position of the cooca coinear fieds. Received February 6, 2001

2 24 S. KRASIĆ THE PROCEDURE OF DEFINING CORRESPONDING IDENTICAL TUFTS This procedure wi be worked on in detais for the most genera situation in this study. Specific situations wi be expained in short, especiay the question why their appication on defining corresponding identica tufts is more convenient than the genera procedure. The procedure of defining corresponding identica tufts is based on choosing a identica perspective tufts, ( 2 ), first, for which the perspective axis is the endessy distant ine, so that their corresponding perspective tufts in the second fied have the perspective axis, i. e. the endessy distant ine of the first fied. In the set of 2 perspective tufts of the second fied, the tufts identica to a of the tufts in the set of 2 perspective identica tufts in the first fied are ooked for. Doing this the given postuate is foowed: the projecting tuft of the first cass ines is defined by three concurrent rays - three ines, which are intersected at the same point, cose up three pairs of suppementary anges (it is necessary for each pair to have one chosen ange, whie the other one is a suppement up to 180 ); three rays, which cose up three anges, define the tuft of the first cass ines; out of three, two anges aways define the third one (which is either addition or subtraction of the first two anges); the tuft of the first cass ines is defined by two anges which are presented with three rays (one of them is mutua); if the anges are equa, then they are both simiar and identica. Fig. 1a. Fig. 1b. α 1 = α α 1 = α 2 α + β = γ α = γ - β β = γ - α 1. GENERAL PROCEDURE There in the P (fig. 2.) fied is 2 of the points which can be tops of the tufts of the first cass ines. The point A is chosen as a top of the tuft, A ( a, b, c), whose rays reciprocay cose up the anges α ( a, b) and β ( b, c). If we repeat the same tuft 2 times, using a the points of the surface as peaks of the tufts, so that the corresponding rays remain reciprocay parae, the tufts wi be perspective ( the perspective axis is the endessy distant ine 1 n of the P fied. There in the P fied 2 corresponding perspective tufts wi intersect the endessy distant ine 1 n in the series 1 n(i II III). The tufts are being ooked for in the P fied, whose rays intersect the endessy distant ine 1 n in the series 1 n(i II III), whie they reciprocay cose up the anges α(a,b) and β(b,c), where the b ray is a mutua one. The tops of a anges α whose rays go through the points I and II make the circumferences k 1 and k 1 (fig. 2) and tops of a the anges β, whose egs go through the points II and III, make circumferences k 2 and k 2. The circumferences k 1 and k 2 intersect

3 Defining Corresponding Identica Tufts in Cooca Genera Coinear Fieds 25 each other at two rea points A 1 and II, which meet both conditions they represent the mutua top for α and β. At first sight it ooks ike there are two points which can be the tops of the identica tufts. But, the A 1 point is the ony top of the identica tuft because the rays a 1, b 1 and c 1 of the (A ) tuft cose up the reciproca anges α (a, b ) and β (b,c ) and they aso have a mutua eg, the b ray, whie they intersect the endessy distant ine 1 n in the series (I II III). The point II is not a top of the identica tuft, because the b ray is not mutua for the anges α and β. Fig. 2. Symmetricay, in reation to the endessy distant ine 1 n, the circumferences k 1 and k 2 (fig. 3.), with the periphera anges α, above the chord I II and β, above the chord II III, intersect each other at the two points A 1 and II. The ony top of the identica tuft (A 1 ) is the A 1 point, whose rintersect the endessy distant ine 1 n, in the series 1n (I II III), cosing up the anges α (a 1,b 1 ) and β (b 1,c 1 ) reciprocay, whie b 1 is the mutua ray for the anges α and β. The A 1 point is orthogonay symmetrica to the A point in reation to the endessy distant ine 1 n. Thous, the P fied contains two tufts, the (A ) and (A 1 ) tuft, which have their own two corresponding identica tufts, ( A ) and ( A 1 ), in the set of ² identica perspective tufts in the P fied. Fig. 3.

4 26 S. KRASIĆ 2. THE SPECIAL PROCEDURE The ( A ) tuft can be presented with three rays of any kind, i. e. there can be used the rays which make more convenient anges-the right ones in this situation. This can produce the foowing variants: - one out of the two anges in the ( A ) tuft is right whie the other one is arbitrary; - both anges in the ( A ) tuft are right (circuar-invountary tuft) One out of the two anges in the ( A ) tuft is right- the other one is arbitrary Genera position of the rays in the ( A ) tuft If the rays in the A tuft ( a, b, c) in the P, fied are presented in the way that the anges paced between the rays are α ( a, b) - as an arbitrary ange, φ ( b, c) - as a right and acute one in reation to n, whie foowing the basic principe for defining tops of corresponding identica tufts, then it is possibe to simpify the constructing procedure which eads to the foowing: - in the P fied the set of ² perspective tufts intesect the endessy distant ine 1 n in the series 1 n (I II III). If the tufts are to be defined as identica to a the tufts in the set of ² identica perspectives in the P fied, then it is necessary to construct the circumference k, (fig. 4.), whose diameter is a segment II III on the endessy distant ine 1 n (incuding a of the tops of a right anges φ in the given identica tuft) and aso circumferences k 1 and k 2, whose chord is a segment I II on the endessy distant ine 1 n (incuding a of the tops of a arbitrary anges α in the expected identica tuft, which are periphera for the given chord). The section of the circumference k and the circumferences k 1 and k 2, the two points, A and A 1 are made, as the potentia tops of the identica Fig. 4. tufts in the set of ² perspective tufts in the P fied. (Fig. 4.) The specia position of the rays in the ( A ) tuft In this situation, the a ray in the A tuft ( a, b, c), is in the arbitrary position in reation to the endessy distant ine n, b is vertica to the endessy distant ine n and, at same time, c it is parae with n. In the ( A ) tuft α ( a, b) is arbitrary, but φ ( b, c) is a right ange. The basic principe for defining tops of corresponding identica tufts has ben foowed in this situation. Since in the P fied there is ony one ray which is vertica to the endessy distant ine 1 n, and that is the main vertica ine ng, (Fig. 5.), which intersects the endessy distant ine through the centre of the copied absoute invoution 1 O, that is the wanted mutua

5 Defining Corresponding Identica Tufts in Cooca Genera Coinear Fieds 27 ray b for both anges α and φ in the identica tuft in the P fied, can ony be the main vertica ine ng = b = b 1. Then, the construction wi remain ony as the circumference section k 1 over the I 1 O chord (whose periphera anges over the chord are the α anges) and the main vertica ine ng = b = b 1. In this way two points A and A 1 are produced and become the tops of the identica tufts in the set of ² perspective in the P fied. (Fig. 5). Fig. 5. The construction of the corresponding identica tufts tops, using the main vertica ine and the ange cosed up atogether with the endessy distant ine. The previousy described consruction can be simpified if the β ange is obvious in the tuft ( A ) ( b, g, m), α ( b, g) and φ ( g, m), (Fig. 6), and which is cosed up with the b ray and the endessy distant ine n, whie the g ray is vertica to the endessy distant ine n. In the set of ² perspective tufts in the P fied, the b ray (Fig. 6) has to form the α ange with the ng = g ray, and then the β ange with the endessy distant ine 1 n, because ng 1 n ( the anges which have one mutua eg and the other one is parae are equa). In this situation it is necessary to determine the point I on the endessy distant ine 1 n which is concurrent to the b ray, and then to construct the β ange, so that the point I becomes a Fig. 6.

6 28 S. KRASIĆ top, whie one eg becomes the endessy distant ine 1 n. The second eg of the β ange, that is the b ray, and the b 1 ray which is symmetrica to the endessy distant ine 1 n, intersect the main vertica ine ng = g = g 1, at the points A and A 1. These two points are the tops of the identica tufts in the set of ² perspective tufts in the P fied. Instead of the circumference above the chord, it is necessary to construct an ange of a corresponding ray with the endessy distant ine, as we as the main vertica ine in the P fied. (Fig. 7) Both anges in the ( A ) tuft are right Fig Genera position of the ray in the ( A ) tuft If the two right anges, φ ( a, b) and φ ( b, c) are peresented in the arbitrary ( A ) tuft in the P fied, the c ray coincides with a, so that such a tuft becomes circuar and invoute.the defining the tops of identica tufts in the P fied demands another pair of orthogona rays in the ( A ) tuft which are e and d, φ ( e, d). If the tuft A ( a, b, c, d) in the P fied is repeated ² times, so that the adequate rays remain parae, a of them wi intersect the fictitious ine 1 n in an absoute invoute series. Their corresponding tufts wi intersect the endessy distant ine 1 n in the P fied in the eiptica invoute series 1 n (I I II II ), (Fig. 8).The tops of the ony two circuar and invoute tufts, (A ) and (A 1 ) in the P fied, identica to the circuar and invoute tuft ( A ) in the P fied, are formed at the section of the circumference k 1, whose diameter is the segment I I, and the section of the circumference k 2, whose diameter is the segment II II. (Fig. 8). Fig The specia ray position in the ( A ) tuft The circuar and invoute tuft ( A ) in the P fid can be given so that one pair of the orthogona rays gets a speci position in reation to the endessy distant ine n. One of the rays, e, is vertica to the endessy distant ine n, whie the other one, d is parae with it.

7 Defining Corresponding Identica Tufts in Cooca Genera Coinear Fieds 29 If the ( A ) tuft is repeated ² times, so that the adequate rays remain parae, then a of them wi intersect the fictitious ine 1 n in an absoute invoute series, whie their corresponding ² tufts in the P fied wi intersect the endessy distant ine in an invoute eiptica series 1 n (I I 1O 2 O ). There is ony one ray which is vertica to the endessy distant ine 1 n in the P fied, where appears e = n g, going through the centre of the copied absoute invoution ı O. At the section of the circumference k, (Fig. 9), whose diameter is the segment I I, and the main vertica ine e = e 1 = n g, the two points, A and A 1 are formed. They are the ony two circuar and invoute tufts (A ) and (A 1 ) in the P fied which have their own corresponding identica circuar and invoute tufts A and A 1 in the P fied. (Fig. 9). Fig. 9. CONCLUSION It appears from the previous that there are ony two tufts, (A ) and ( A 1 ) in the P fied that have their own two corresponding tufts ( A ) and ( A 1 ) in the P fied. Projecting correspondence is possibe since the corresponding rays intersect one another on the corresponding ines, whie the equaity is gained by having the identica anges between the corresponding rays. In any way of choosing the rays in the tufts (A ) and ( A ) in the first fied witht the arbitrariy taken anges, which can aso be the right 1 ones, their two corresponding tufts ( A ) and ( A 1 ) in the second fied wi have equa anges between the rays, that is, they're going to be identica. If the accepted anges are right, then the tufts are circuar and invoute with the tops as focuses F,F 1, F, F 1, which are La-Guerr s points of copied absoute invoutions on the endessy distant ine, appearing in pairs, so that this foows: F A, F 1 A 1, F A and 1 F 1 A. A in a, there are two corresponding pairs of identica tufts in cooca genera coinear fieds and they have focuses as tops. Constructive procedure for defining focuses used up to now, incudes construction of circumferences over the points of circuar and invoute series on the endessy distant ine which is corresponding to the absoute invoute series on the endessy distant ine. This

8 30 S. KRASIĆ study produces the simper way of construction for defining focuses. It is based on the construction of the main vertica ine and the ange formed with the endessy distant ine in the second fied and which the corresponding ray in the first fied with its own endessy distant ine coses up. This construction is given in detais in the chapter. REFERENCES 1. Niče V. Deskriptivna geometrija. Škoska knjiga, Zagreb, Sbutega V. Sintetička geometrija. Gradjevinski fakutet, Beograd, Jovanović A. Magistarski rad. Arhitektonski fakutet, Beograd IDENTIČNI PRIDRUŽENI PRAMENOVI PRAVIH KAO INVARIJANTE U KOLOKALNIM OPŠTE-KOLINEARNIM POLJIMA Sonja Krasić Kookana koinearna poja se iz opšteg mogu dovesti u perspektivni poožaj, u kome je postupak presikavanja pojednostavjen. Ako postoje medju projektivnim, identični pramenovi pravih koji su pri tom pridruženi, njihovim pokapanjem poja se dovode u perspektivni poožaj. Postupak odredjivanja pridruženih identičnih pramenova pravih sastoji se u tome da se u skupu od 2 perspektivnih pramenova u jednom poju (čija je osa perspektiviteta nedogednica), pronadju oni koji su identični sa svim pramenovima u skupu od 2 perspektivnih identičnih pramenova u drugom poju (čija je osa perspektiviteta beskonačno daeka prava). Pri tom se poazi od opšteg sučaja, proizvojno uzetih zraka u pramenovima u jednom poju. Zatim se postupak odredjivanja pridruženih identičnih pramenova pojednostavjuje uvodjenjem specijano uzetih zraka u pramenovima u prvom poju. Zbog svojih osobina koje zavise samo od projektiviteta zadatog četvorkom jednoznačno pridruženih tačaka (pravih), pridruženi identični pramenovi pravih se ubrajaju u invarijante opšte-koinearnih i perspektivno-koinearnih poja. Zakjučak je da u opštekoinearnim pojima postoje dva pridružena para identičnih pramenova.