Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Subdivision is an important and e cient tool for rendering smooth curves and surfaces in computer graphics, by repeatedly applying a subdivision (re ning) scheme to a given set of points. In the literature, attention has been mostly restricted to developing binary subdivision schemes. The primary emphasis of this thesis is on ternary subdivision, and in particular on the interpolatory case. We will derive a symmetric ternary interpolatory subdivision scheme for the rendering of curves, satisfying analogous properties to the Dubuc-Deslauriers binary scheme. Explicit construction methods, as well as a corresponding convergence analysis, will be presented. Graphical illustrations of the results will also be provided. / AFRIKAANSE OPSOMMING: Subdivisie bied 'n belangrike en doeltre ende metode om gladde krommes en oppervlakke in rekenaargra ka te genereer. Hierdie metode behels dat 'n subdivisieskema (of verfyningskema) herhaaldelik toegepas word op 'n gegewe versameling punte. In die literatuur word daar hoofsaaklik gefokus op die ont- wikkeling van bin^ere subdivisieskemas. In hierdie tesis word die klem gel^e op tern^ere subdivisieskemas, en in die besonder op interpolerende skemas. Ons sal 'n simmetriese tern^ere interpolerende subdivisieskema, wat analo e eienskappe as di e van die Dubuc-Deslauriers bin^ere skema bevredig, ontwikkel, om krom- mes te lewer. Eksplisiete konstruksiemetodes en ooreenkomstige konvergensie- analise, asook gra ese illustrasies van die resultate, sal getoon word.