Vol.2, No 2, 2000
pp. 117-129
UDC 514.18:681.3.06(045)
DETERMINATION OF INTERSECTING CURVE BETWEEN
TWO SURFACES OF REVOLUTION WITH INTERSECTING AXES BY USE OF AUXILIARY SPHERES
Ratko Obradović
Faculty of Technical Sciences,
Institute for Mathematics and Physics in Engineering,
Trg Dositeja Obradovića 6,
21121, Novi Sad, Serbia, Yugoslavia, Email: obrad_r@uns.ns.ac.yu
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).
Key words: Surface of Revolution,
Auxiliary Sphere, Descriptive Geometry, Computer Graphics.
ODREĐIVANJE PRESEKA DVEJU ROTACIONIH
POVRŠI
ČIJE SE OSE SEKU KORIŠĆENJEM POMOĆNIH LOPTI
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).
Ključne reči: rotaciona površ, pomoćna
lopta, deskriptivna geometrija, kompjuterska grafika.
