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2000, vol. 2, br. 2, str. 117-129
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Određivanje preseka dveju rotacionih površi čije se ose seku korišćenjem pomoćnih lopti
Determination of intersecting curve between two surfaces of revolution with intersecting axes by use of auxiliary spheres
Projekat: Projekat Ministarstva nauke Republike Srbije, br. 1819: Synthesis and development of geometric knowledge for advancement of visualization and representation of spatial configurations
Sažetak
Metod pomoćnih lopti se u deskriptivnoj geometriji koristi kod određivanja preseka dveju rotacionih površi čije se ose seku (Dovniković 1994; Anagnosti 1984). Naime, ako se u tačku preseka osa rotacija dveju površi postavi centar svih pomoćnih lopti, tada svaka pomoćna lopta u opštem slučaju seče svaku rotacionu površ po k krugova-paralela čije su ravni upravne na osu rotacione površi (k=2m, jer je krug drugog reda a kriva koja obrazuje rotacionu površ je reda m). Za slučaj kada lopta seče obe površi po dvema paralelama, parovi paralela se seku u najviše 8 tačaka i te presečne tačke paralela su tačke prostorne presečne krive dveju površi za posmatranu pomoćnu loptu. Menjanjem prečnika pomoćnih lopti dobijaju se nove paralele i u njihovom preseku nove tačke prostorne presečne krive (Obradović 2000). Prikazanu ideju moguće je matematički formulisati i oformiti postupak u kojem neće biti važno kojim se redosledom spajaju presečne tačke za sve pomoćne lopte, jer će se procedura realizovati pomoću računara, pa će se, sa dovoljno velikim brojem pomoćnih lopti, problem spajanja presečnih tačaka izbeći (Obradović 2000).
Abstract
A descriptive geometrical method of auxiliary spheres has been used for the determination of intersecting curve between two surfaces of revolution with intersecting axes of surfaces (Dovniković 1994, Anagnosti 1984). Namely, if the centre of all auxiliary spheres is put on intersecting points between axes of surfaces then in general each auxiliary sphere intersects each surface of revolution on k circles-parallels (k=2m because circle is second order's curve and the starting curve of surface of revolution is order m). These circles lie in planes orthographic on axis of the surface of revolution. In case when sphere intersects both surfaces on two circles then the pair of circles are intersected at most 8 points. These intersecting points of circles are points of intersecting space curve between two surfaces for this sphere. New circles and new points of intersecting curve have been determined by radius changing of auxiliary spheres (Obradović 2000). It is possible to make the mathematical form and procedure for this idea and in this case the sequence for the connection of intersecting curve points will not be important because procedure will be realized by computer. With a sufficiently large number of auxiliary spheres it will be possible to avoid connecting problem of intersecting curve points (Obradović 2000).
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