Vol.9, No 2, 2011 pp. 269 - 275
UDC 744.42:514.18:514.144=111
DOI: 10.2298/FUACE1102269K
GRAPHIC REPRESENTATION OF A TRIAXIAL ELLIPSOID BY MEANS OF A SPHERE IN GENERAL COLLINEAR SPACES
Sonja Krasić*, Biserka Marković
University of Nis, The Faculty of Architecture and Civil Engineering, Serbia
E-mail: *sonja.krasic@gaf.ni.ac.rs
For graphic representation of the projective creations, such as the quadrics (II degree surfaces) in projective, general collinear spaces, it is necessary to firstly determine the characteristic parameters, such as: vanishing planes, axes and centers of space. An absolute conic of a space is an imaginary conic, residing in the infinitely distant plane of that space. The common elements of the absolute conic and infinitely distant conic of a quadric in the infinitely distant plane of that space are the autopolar triangle and two double straight lines which are always real and it is necessary to use the common elements of their associated pair of conics in the vanishing plane of the associated space. The quadric axes are passing through the apices of the autopolar triangle, and they are important for graphic representation of the quadrics. In order to map a sphere in the first space into the triaxial ellipsoid in the second space, it is necessary to select a sphere so that its center is not on the axis of that space and that it intersects the vanishing plane of the second space along the imaginary circumference, which is in general position with the figure of the absolute conic of the second space (the associated pair of conics in the vanishing plane).
Key words: general collinear spaces, the absolute conic, sphere, triaxial ellipsoid.
GRAFIČKO PREDSTAVLJANJE TROOSNOG ELIPSOIDA POMOĆU SFERE U OPŠTE KOLINEARNIM PROSTORIMA
Za grafičko predstavljanje projektivnih tvorevina u koje spadajui kvadrike (površi II stepena) u projektivnim, opšte kolinearnim prostorima, potrebno je najpre odrediti karakteristične parametre i to: nedogledne ravni, ose i centre prostora. Apsolutna konika nekog prostora je imaginarna konika i nalazi se u beskonačno dalekoj ravni tog prostora. Zajednički elementi apsolutne konike i beskonačno daleke konike neke kvadrike su autopolarni trougao i dve dvostruke prave koji su uvek realni u tom prostoru i potrebno je koristiti zajedničke elemente njihovog pridruženog para konika u nedoglednoj ravni pridruženog prostora. Kroz temena autopolarnog trougla prolaze ose kvadrike koje su važne za grafičko predstavljanje kvadrika.Da bi se sfera u prvom prostoru, preslikala u troosni elipsoid u drugom prostoru, potrebno je sferu izabrati tako da joj središte nije na osi tog prostora i da nedoglednu ravan seče po imaginarnoj kružnici, koja je sa slikom apsolutne konike, u opštem položaju (pridruženi par konika u nedoglednoj ravni) .
Ključne reči: opšte kolinearni prostori, apsolutna konika, sfera, troosni elipsoid