ANALYSIS OF MAPPING OF GENERAL II DEGREE SURFACES IN COLLINEAR SPACES UDC (045)=111

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1 FACTA UNIVERSITATIS Series: Architecture and Civil Engineering Vol. 8, N o 3, 2010, pp DOI: /FUACE K ANALYSIS OF MAPPING OF GENERAL II DEGREE SURFACES IN COLLINEAR SPACES UDC (045)=111 Sonja Krasić *, Biserka Marković University of Niš, The Faculty of Civil Engineering and Architecture, Serbia * sonjak@gaf.ni.ac.rs Abstract. Mapping of projective creations which includes the II degree surfaces in the projective, general collinear spaces is complex. In order to simplify it, firstly the characteristic parameters must be constructively determined: vanishing planes, axes and centers of spaces. All II degree surfaces are mapped using the common elements of absolute conic and infinitely distant conic of quadrics in the infinitely distant plane of space, which provide the determination of parameters of any surface of II degree. The common elements of their associated pair of conics in vanishing plane of space are used. The paper analyzed the conditions of choice of general surface of II degree in the first space to be mapped into the respective general surface of II degree in the second collinear space. The mapping is biunivocal. A sphere is chosen in the first space, and it was analyzed how it should be placed in respect to the characteristic parameters of the space, so that it would be mapped in rotating or triaxial general surfaces of II degree in the second space. Key words: collinear spaces, general surface of II degree, characteristic parameters of space, rotating surfaces, triaxial surfaces 1. INTRODUCTION Two projective, collinear spaces ( 1 and 2 ) are in a general case given with five pairs of biunivocally associated points (A 1 B 1 C 1 D 1 E 1 and A 2 B 2 C 2 D 2 E 2 ). In order to simplify mapping of projective creations including surfaces of II degree in these spaces, firstly the characteristic parameters must be constructively determined, such as:1. vanishing planes (N 1 and M 2 ), which are associated with two infinitely distant planes of space (N 2 and M 1 ); 2. axes of space (o 1 and o 2 ), only two associated straight lines perpendicular to the vanishing planes; 3. space centers (W 1 andx 2 ), axis piercing points through the vanishing planes. Each plane, even an infinitely distant one, intersects the general surface of II degree on the curve of II degree, which may be real, imaginary or imaginary degenerated into two conjugate imaginary straight lines. All surfaces of II Received June 15, 2010

2 318 S. KRASIĆ, B. MARKOVIĆ degree are mapped using the common elements of absolute conic and infinitely distant conic of quadric, which provide the determination of axes, circular intersections and circumferential points of any surface of II degree. "Each plane in space has its own infinity distant straight line which contains the absolute points of that plane. Geometric position of absolute points in the infinitely distant plane of the space is absolute conic of space. This imaginary curve of II degree intersects every straight line in infinitely distant plane in a pair of conjugate imaginary points". [6] Since the absolute and infinitely distant conic are situated in the infinitely distant plane of one space and cannot be represented graphically, the associated pair of conics in a vanishing plane of another space is used, that is, their common elements. Absolute conic a I1 associated with the imaginary circle a I2, which has a real representation a Z2 (figure of absolute conic). Infinitely distant conic of quadrics k 1 has its own associated conic k 2, which was obtained by the intersection of the associated quadric with the vanishing plane. The paper analyzed the conditions that must be satisfied by the general surface of II degree in the first space so that it could be mapped into the respective general surface of II degree in the second collinear space. A sphere is chosen for the surface in the first space. Therefore according to what its position in respect to the characteristic parameters of the first space is, it will be mapped either into the rotational or into the triaxial general surface. According to the infinitely distant conic of general surface of II degree (real, imaginary or degenerated imaginary), a sphere will be mapped into a ellipsoid, a hyperboloid of two sheets, a paraboloid. Mapping is biunivocal. 2. DETERMINATION OF THE CHARASTERISTIC PARAMETERS AND FIGURES OF ABSOLUTE CONICS IN GENERAL COLLINEAR SPACES The constructive procedure for determination of characteristic parameters of spaces (fig. 1) was given in more detail in other papers [4], and that for determination of figures of absolute conics (fig. 2) in the paper [3]. 3. GENERAL SURFACES OF II DEGREE General surfaces of II degree, formed as a product of projective fundamental creations of 2 nd kind. The paper considered only real general surface of II degree. Infinitely distant conic of quadrics these may be real (hyperboloid of two sheets), imaginary (ellipsoid) and imaginary degenerated into two conjugate imaginary straight lines (paraboloid). A sphere is a special type of ellipsoid, whose infinitely distant conic is absolute conic. In order to map general surfaces of II degree, a sphere has been chosen in the first space, whose intersections with any plane are the circumferences, which may be real, imaginary or imaginary degenerated, which will be used for the further analysis. For the purpose of determining axes, circular intersections and circumferential points of associated general surface of II degree, in the second space, we used common elements of intersecting conic of the sphere with the vanishing plane and the figure of absolute conic in the first space. These are common autopolar triangle and two real double straight lines, lying on the conjugate imaginary intersecting points of these two conics in vanishing plane in the first space.

3 Analysis of Mappingo General II Degree Surfaces in Collinear Spaces 319 Fig. 1 Determination of the characteristic parameters in general collinear spaces Fig. 2 Determination of figures of absolute conics in general collinear spaces

4 320 S. KRASIĆ, B. MARKOVIĆ Through the apexes of the common autopolar triangle of their associated conics in infinitely distant plane and through the center pass the axes of the mapped general surface in second space. In the general case the surfaces of II degree have three axes. The center of a sphere in the first space maps into the pole in respect to the vanishing plane of the associated surface in the second space. The pole of the sphere in respect to the vanishing plane in the first space maps into the center of the associated surface in the second space. Real double straight lines determine the two systems of parallel planes that will intersect the associated surface along circumferences circles. Four planes from the two systems of parallel planes touch the surface in circular points. 4. MAPPING A SPHERE INTO ROTATING GENERAL SURFACES OF II DEGREE "In general rotating surfaces of II degree, out of the three points of common autopolar triangle, of the two conics in the vanishing plane, intersecting conic of sphere k 2 with vanishing plane M 2, and the figure of absolute conic a I2, two are infinitely distant, R 2 and Q 2, and one is in finiteness P 2. "Each pair of involutory associated points on the infinitely distant straight line of vanishing plane M 2, induced by two conics, k 2and a I2, with the point P 2 constitute a common autopolar triangle, which means that there is 1 of such triangles". [6] General rotating surfaces of II degree, have a single main axis, with the center of that surface lying on it. The main axis of the associated general surfaces in the second space is the axis of that space. [2] In order to map a sphere s 2 in the space 2 (fig. 3), into a rotating ellipsoid s 1 in space 1, it is necessary to select a sphere in such a way that it intersects the vanishing plane M 2 in the first space, along the imaginary circumference k I2, whose real representative is the circumference k Z2, which is concentric with a figure a I2, of absolute conic a I1 in the first space (fig. 3). Since the imaginary centers of circumferences k I2 and a I2, whose real representative are the circumferences k Z2 and a Z2, coincide with each other and with center W 2 of space 2, that point is the point P 2 in finiteness of common autopolar triangle of these two conics, Z 2 = W 2 = P 2. Point W 2 of space 2 is associated with the point W 1 of space 1, implying that the main axis of the rotational ellipsoid s 1 in space 1, is axis o 1 of space 1, because its infinitely distant point is W 1, W 1 = P 1. There is only one system of parallel planes that intersects a rotating ellipsoid by circumferences and only two real circumferential points in finiteness, in which the planes touches the ellipsoid. All other points are elliptical. [2] In order to map a sphere s 2 in space 2 (fig. 4), into a rotating hyperboloid of two sheets s 1 in space 1, it is necessary to select a sphere in such a way that it intersect the vanishing plane M 2 in the first space, along the real circumference k 2, which is concentric with a figure of absolute conic a I2, its real representative a Z2, (fig. 4). Everything else related to the axis, circular cross-sections and circular points of rotating hyperboloid of two sheets, valid as the rotating ellipsoid.

5 Analysis of Mappingo General II Degree Surfaces in Collinear Spaces Fig. 3 Mapping a sphere into rotating ellipsoid Fig. 4 Mapping sphere into rotating hyperboloid of two sheets 321

6 322 S. KRASIĆ, B. MARKOVIĆ In order to map a sphere s 2 in space 2 (fig. 5), into a rotating paraboloid s 1 in space 1, it is necessary to select a sphere in such a way that it intersects the vanishing plane M 2 in the first space, along the degenerated imaginary conic k I2, which degenerates into two conjugate imaginary lines. This means that a sphere touches the vanishing plane in that space. Of the three points of common autopolar triangle for the circumerence a Z2 and two conjugate imaginary straight lines Z 2 (b 2,b 2,c 2,c 2 ) in the vanishing plane M 2, two are infinitely distant, R 2 and Q 2, and one is in finiteness P 2, as in the other rotating surfaces. Everything else related to the axes and circular cross-sections of rotating paraboloid, is the same as in a rotating ellipsoid, apart from one circumferential point is in finiteness, and other is infinitely distant. [2] Fig. 5 Mapping a sphere into a rotating paraboloid 4. MAPPING A SPHERE INTO A TRIAXIAL GENERAL SURFACES OF II DEGREE In triaxial general surfaces of II degree, common elements of the absolute conic of space and infinitely distant conic of quadrics ensures that these surfaces have three axes and two systems of parallel planes to intersect them along circumferences. Four planes of the two systems of planes touch general surfaces in the circumferential points, which means that there are four circular points. Of the three points of common autopolar triangle of their associated pair of conics a I2 and k 2 in the vanishing plane M 2, two of them are in the finiteness P 2 and R 2, and one is infinitely distant Q 2. The centers of the two conics in

7 Analysis of Mappingo General II Degree Surfaces in Collinear Spaces 323 vanishing plane M 2 do not coincide, or they are in general position. This means that the center of a sphere in the first space should not be on the axis of that space. [2] In order to map a sphere s 2 in space 2 (fig.6), into a triaxial ellipsoid s 1 in space 1, it is necessary to select a sphere in such a way that it intersect the vanishing plane M 2 in the first space, along the imaginary circumference k I2, whose the real representative is the circumference k Z2, which is in general position with a figure a I2 of absolute conic a I1. This means that the sphere has no common real points with the vanishing plane M 2. The apexes of the common autopolar triangle in the infinitely distant plane of the space 1 with center K 1 determine the three real axis of triaxial ellipsoid. Double straight lines d 1 and d 2 determine two systems of parallel planes that intersect the triaxial ellipsoid along circumferences. There are four real circumferential points in finiteness. All other points of triaxial ellipsoid are elliptical. [2] Fig. 6 Mapping a sphere into a triaxial ellipsoid In order to map a sphere s 2 in space 2 (fig.7), into a triaxial (elliptical) hyperboloid of two sheets in space 1, it is necessary to select a sphere in such a way that it intersects the vanishing plane M 2 in the first space,along the real circumference k 2, which is in general position with a figure a I2 of absolute conic a I1. Points P 1, Q 1 and R 1 of common autopolar triangle in the infinite distant plane of space 1 and the center K 1 determine the three axes of triaxial hyperboloid of two sheets. One of the axis is real, and two of them are conjugate imaginary ones. Everything that applies to the circumferential

8 324 S. KRASIĆ, B. MARKOVIĆ intersections and circumferential points of triaxial ellipsoid also apply for triaxial (elliptical) hyperboloid of two sheets. [2] Fig. 7 Mapping a sphere into a triaxial (elliptical) hyperboloid of two sheets In order to map a sphere s 2 in space 2 (fig.8), into a triaxial (elliptical) paraboloid in space 1, it is necessary to select a sphere in such a way that it intersects the vanishing plane M 2 in the first space, along the imaginary degenerated conic k I2, which degenerats into two conjugate imaginary straight lines, whose intersecting point is Z 2 (the sphere touches the vanishing plane). The point W 2 which is the center of circumference a Z2, which is the real representative of the conic a I2, does not coincide with the point Z 2, which means that center of the sphere is not on the axis of the space 2. Three-axes of triaxial paraboloid are determined by three points P 1, Q 1 and R 1 of common autopolar triangle in infinitely distant plane of space 1 and the center K 1. Everything that applies to the circular cross-sections and circular points of triaxial ellipsoid also applies to triaxial (elliptical) paraboloid, only difference being two circular points which are real in finiteness, and other two are real in infinitely distant plane. [2]

9 Analysis of Mappingo General II Degree Surfaces in Collinear Spaces 325 Fig. 8 Mapping a sphere into a triaxial (elliptical) paraboloid 5. MAPPING A SPHERE INTO A SPHERE This is a special case of mapping of general surfaces of II degree, since the infinitely distant conic of spheres is the absolute conic. In order to map a sphere s 2 in space 2 (fig. 9) into a sphere in space 1 it is necessary to select a sphere in such a way that it intersect the vanishing plane M 2 in the first space, along the figure of absolute conic a I2, an imaginary circumference whose the real representative is the circumference a Z2. Centers of figures of absolute conics in vanishing planes coincide with the centers of spaces, therefore the centers of the spheres, of the elliptic pencil of spheres, which maps into an elliptic pencil of spheres, must be on the axes of the spaces. All points on a sphere are the circumferential points. [2]

10 326 S. KRASIĆ, B. MARKOVIĆ Fig. 9. Mapping a sphere into a sphere 7. CONCLUSION The paper concludes that general surfaces of II degree in the collinear spaces will always be mapped into general surfaces of II degree. In order to make the mapped surfaces of II degree in the second space: 1. rotational, centers of the surface in the first space must be on the axis of that space (in this way, providing the main axis of rotational surface); 2. triaxial (elliptical), the center of the surface in the first space should not be on the axis of that space. For rotational and triaxial surfaces, in order to make the mapped surface of II degree in the second space: 1. ellipsoid, surface intersection with vanishing plane in the first space, is an imaginary circumference; 2. hyperboloid of two sheets, surface intersection with vanishing plane in the first space is real circumference; 3. paraboloid, surface intersection with vanishing plane in the first space is degenerated imaginary circumference, which degenerates to two conjugate imaginary straight lines. Should the mapped surface of II degree in the second space be a sphere, surface intersection with vanishing plane in the first space must be the figure of absolute conic. REFERENCES 1. Jovanovic A.: Metric and position invariants of collinear spaces, PhD Thesis, Faculty of Architecture, Beograd, Krasic S.: Association quadrics with absolute conic of general collinear spaces, PhD Thesis, Faculty of Civil engineering and Architecture, Niš, 2007.

11 Analysis of Mappingo General II Degree Surfaces in Collinear Spaces Krasic S., Nikolic V.: Constructive procedure for determination of absolute conic figure in general collinear spaces, Filomat, Volume 23, Number 2, University of Nis, Nis, pg , June Krasic S., Markovic M.: Determination of the characteristic parameters in the general collinear spaces in the general case, Facta Universitatis, Series Architecture and Civil Engineering, Vol. 3, No.2, University of Nis, Nis, pg , Markovic M.: Imaginary curves and surfaces of II degree, PhD Thesis, Faculty of Architecture, Beograd, Niče V.: Introduction to Synthetic Geometry, Školska knjiga, Zagreb, ANALIZA PRESLIKAVANJA OPŠTIH POVRŠI II STEPENA U KOLINEARNIM PROSTORIMA Sonja Krasić, Biserka Marković Preslikavanje svih projektivnih tvorevina u koje ubrajamo i površi II stepena u projektivnim, opšte kolinearnim prostorima je složeno. Da bi se pojednostavilo, najpre se moraju konstruktivno odrediti karakteristični parametri: nedogledne ravni, ose i centri prostora. Sve površi II stepena preslikavaju se pomoću zajedničkih elemenata apsolutne konike i beskonačno daleke konike kvadrike u beskonačno dalekoj ravni prostora, koji obezbeđuju određivanje parametara bilo koje površi II stepena. Koriste se zajednički elementi njihovog pridruženog para konika u nedoglednoj ravni prostora. U radu su analizirani uslovi izbora opštih površi II stepena u prvom prostoru da bi se preslikale u odgovarajuće opšte površi II stepena u drugom kolinearnom prostoru. Preslikavanje je obostrano jednoznačno. U prvom prostoru birana je sfera i analizirano je kako je treba postaviti u odnosu na karakteristične parametre prostora, da bi se preslikala u rotacione (obrtne) ili troosne opšte površi II stepena u drugom prostoru. Ključne reči: kolinearni prostori, opšte površi II stepena, karakteristični parametri prostora, rotacione površi, troosne površi