Vol.2, No 3, 2001 pp. 219-230
UDC 514.7 (045)
COMPARISON OF THE ANALYTIC AND SYNTHETIC METHODS FOR TRANSFORMATION OF THE NTH ORDER CURVES INTO THE 2NTH ORDER CURVES
Biserka Marković
Faculty of Civil Engineering and Architecture, Niš, Aleksandra Medvedeva 14, Serbia
Abstract. This paper analyses the quadratic transformation, as known in the literature, which is effected by the analytic geometry procedures in trilinear coordinate system, as well as the graphic construction based on it. Also, the square transformation, based on the methods and the graphic procedures of the synthetic geometry is shown, and it has been realised in the pencils of the polar fields, set by a pair of fundamental conics, with the autopolar triangle. By the comparison of these two methods, the similiraties and differences are presented, and the invariants of both methods were established. The paper shows that the analytic and synthetic method, though using different means and procedures, are mutually intertwined, complemented and congruent, and strive towards the common goal:  obtaining the variety of different families and forms of curves.
Key words: analytic-synthetic geometry; trilinear coordinate system – autopolar triangle, square transformation, form of curves, pencil of polar fields.
POREDJENJE ANALITIČKE I SINTETIČKE METODE
ZA TRANSFORAMACIJU KRIVIH LINIJA N. REDA U KRIVE 2N. REDA
U ovom radu analizurana je kvadratna transformacija, poznata iz literaure, ostvarena postupcima analitičke geometrije u trilinarnom koordinatnom sistemu i na njoj zasnovana grafička konstrukcija. Isto tako, prikazana je kvadratna transformacija, utemeljena na metodama i grafičkim postupcima sintetičke geometrije, realizovana u pramenu polarnih polja, zadatih parom degenerisanih temeljnih konika, sa autopolarnim trouglom. Poredjenjem ove dve metode istaknute su sličnosti i razlike i ustanovljene invarijante obe metode. Pokazano je da se analitička i sintetička metoda, služeći se različitim sredstvima i postupcima, medjusobno prožimaju, dopnjavaju i preklapaju, na zajedničkom cilju: dobijanju obilja različitih familija i formi krivih linija.