Prodorne krivulje ploha

U radu smo se ograničili na proučavanje prodora valjaka, stožaca i sfera. Najprije smo razmotrili koje krivulje dobijemo kao presjek tih ploha ravninom, a nakon toga kakve krivulje dobijemo u prodoru spomenutih ploha. Upoznali smo metode koje koristimo pri konstrukciji prodornih krivulja dviju ploha u Mongeovoj projekciji. Prva metoda je metoda presjeka s pramenom ravnina. Pomoćne ravnine mogu, ali i ne moraju biti međusobno paralelne. Presjek jedne plohe i pomoćne ravnine je neka krivulja, presjek druge plohe i iste pomoćne ravnine je neka krivulja. Sjecišta tih dviju presječnih krivulja točke su prodorne krivulje. Postupak se ponavlja dok ne dobijemo dovoljan broj točaka koje na kraju spojimo. Druga metoda je metoda presjeka s koncentričnim sferama. Za njezinu primjenu trebaju biti zadovoljena dva uvjeta: plohe trebaju biti rotacijske, osi ploha se trebaju sjeći. Sjecište osi ploha središte je pomoćnih sfera i zbog toga govorimo o koncentričnim sferama. Ako je dan prodor ploha i zadovoljena su oba uvjeta za rad s metodom, sjecišta prodornih krivulja jedne plohe i pomoćne sfere te prodornih krivulja druge plohe i te iste pomoćne sfere daju točke tražene prodorne krivulje ploha. Postupak ponavljamo uzimajući neku drugu sferu koja s plohama ima zajedničke točke. Na kraju rada spomenuta je i tangenta prodorne krivulje. Tangenta u nekoj točki prodorne krivulje presječnica je tangencijalnih ravnina ploha u toj točki.In this thesis, we focus on studying the penetration of cylinders, cones and spheres. To begin with, we have considered which curve is the result of a cross section of these three surfaces with a plane, and after that which curves we get as a result of penetration of the surfaces. We familiarize with the methods used to construct penetrating curves of two of surfaces in Monge’s projection. The first method is a cross section method with a streak of plane. Auxiliary planes could, but don’t have to, be parallel. A cross section of a surface and an auxiliary plane is a curve, whereas a cross section of another surface and the same auxiliary plane is another curve. Intersection of these two curves are points of penetrating curve. The process is repeated until there are enough points to merge them in the end. The second method is a cross section method with concentric spheres. To apply this method, two conditions have to be met: surfaces need to be rotational, and axes of surfaces have to intersect. The intersection of axes is the center of auxiliary spheres and that is the reason why we are talking about concentric spheres. If the penetration of surfaces is known and both conditions to work with the method are met, the intersections of penetrating curves one of a surface and auxiliary sphere, and penetrating curves of another surface and the same auxiliary sphere give points of looked-for penetrating curve of the surfaces. The process is repeated by choosing another sphere which shares some common points with the surfaces. In the end of the thesis, the tangent of penetrating curve is mentioned. The tangent in some point of penetrating curve is a transversal of tangential plane of surfaces in the mentioned point