Krivulje nastale razmotavanjem valjaka i stožaca

master's thesis

• Author: Barbara Sklepić
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Zlatko Drmač (član povjerenstva)
• Committee member: Matija Kazalicki (član povjerenstva)
• Committee member: Sanja Varošanec (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-07-17, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Cilindrične i konusne plohe primjeri su razvojnih pravčastih ploha. Razmotavanjem takvih ploha u ravninu, od krivulja koje leže na tim plohama dobivaju se ravninske krivulje koje su zanimljive ne samo geometrijski, nego i u praktičnim primjenama. U ovom radu izložene su neke metode opisivanja i brojni primjeri krivulja dobivenih razmotavanjem plašteva valjaka i stožaca. Krivulja na valjku može nastati presijecanjem ravninom ili drugim valjkom, odnosno bušenjem cilindričnom bušilicom, pri čemu se varira promjer bušilice i njezina udaljenost od osi valjka. Tipična krivulja dobivena razmatanjem presječne krivulje dvaju cilindara ima oblik \(u(x,z) = p(r \sin{\frac{x}{r}}, z)\) pri čemu je \(p(t,z) = 0\) jednadžba profilne krivulje presijecajućeg cilindra. Promatra se i inverzni problem, kakva se krivulja dobiva namatanjem na valjak neke ravninske krivulje, primjerice kružnice, ovisno o radijusu presjeka kružnog valjka. Ovakvi primjeri često se pojavljuju u primjenama u graditeljstvu i strojarstvu. Kod proučavanja krivulja nastalih razmatanjem konusnih ploha prikladno je primijeniti polarni koordinatni sustav te tzv. stropnu projekciju i kišobransku transformaciju. Osim općenitih krivulja na stošcu, posebno su razmotrene krivulje dobivene tako da se konika s jednog stošca namota na drugi stošac ili razmota u ravninu. Razmotavanjem u ravninu nastaju krivulje koje nazivamo generaliziranim konikama, budući da im je polarna jednadžba oblika \(R(\theta)=\frac{R_0}{1+ \lambda \sin {(k \theta)}}\). Na kraju su prikazani neki od najzanimljivijih primjera generaliziranih konika.

Abstract (english)

Cylindrical and conical surfaces are examples of developable ruled surfaces. By unwrapping such surfaces onto a plane, planar curves are obtained from curves lying on these surfaces which are interesting not only geometrically, but also in practical applications. This paper presents some methods of description and numerous examples of curves obtained by unwrapping lateral surfaces of cones and cylinders. A curve on the cylinder can be formed by cutting by a plane or another cylinder, i.e. drilling with a cylindrical drill, where drill diameter and its distance from the axis of the cylinder are varying. The typical curve obtained by unwrapping the intersection curve of the two cylinders is given by the equation of the form \(u(x,z) = p(r \sin{\frac{x}{r}}, z)\) where \(p(t,z) = 0\) is the equation of the profile curve of two intersecting cylinder. An inverse problem is also observed, namely, what kind of curve is obtained by wrapping onto a cylinder, such as a circle, on a cylinder curve, depending on the radius of the circular cylinder intersection. Such examples often appear in applications in construction and mechanical engineering. When studying curves formed by unwrapping of conical surfaces, it is appropriate to apply a polar coordinate system and so-called ceiling projection and umbrella transformation. In addition to general curves on the cone, curves obtained so that a conic from one cone is wrapped onto another cone or unwrapped onto a plane have also been taken into consideration. Curves obtained by unwrapping onto a plane are shown to have polar equation of the form \(R(\theta)=\frac{R_0}{1+ \lambda \sin {(k \theta)}}\) and therefore are called generalized conics. At the end, some of the most interesting examples of generalized conics are shown.

• Keywords: pravčasta ploha; ravninska krivulja; valjak; stožac; polarni koordinatni sustav; stropna projekcija; kišobranska transformacija; generalizirane konike
• Keywords (english): ruled surface; planar curve; cone; cylinder; polar coordinate system; ceiling projection; umbrella transformation; generalized conics
• Language: croatian
• URN:NBN: urn:nbn:hr:217:145920
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2018-03-15 12:50:49