Implementation of the Cavalieri Integral

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dc.contributor.advisorGrobler, Trienkoen_ZA
dc.contributor.authorvan Zyl, Christoffen_ZA
dc.contributor.other Stellenbosch University. Faculty of Science. Dept. of Computer Science.en_ZA
dc.date.accessioned2023-02-19T18:12:08Zen_ZA
dc.date.accessioned2023-05-18T07:00:35Zen_ZA
dc.date.available2023-02-19T18:12:08Zen_ZA
dc.date.available2023-05-18T07:00:35Zen_ZA
dc.date.issued2023-02en_ZA
dc.descriptionThesis (MSc)--Stellenbosch University, 2023.en_ZA
dc.description.abstractENGLISH ABSTRACT: Cavalieri Integration in R n presents a novel visualization mechanism for weighted integration and challenges the notion of strictly rectangular integration strips. It does so by concealing the integrator inside the boundary curves of the integral. This paper investigates the Cavalieri integral as a superset of Riemann-integration in R n−1 , whereby the integral is defined by a translational region in R n−1 , which uniquely defines the integrand, integrator and integration region. In R 2 , this refined translational region definition allows for the visualization of Riemann-Stieltjes integrals along with other forms of weighted integration such as the Riemann–Liouville fractional integral and convolution operator. Programmatic implementation of such visualizations and computation of integral values are also investigated and relies on knowledge relating to numeric integration, algorithmic differentiation and numeric root finding. For the R 3 case, such visualizations over polygonal regions requires a mechanism for the triangulation of a set of nested polygons and transformations which allow for the use of repeated integration to solve the integration value over the produced triangular regions using standard 1-dimensional integration routines. en_ZA
dc.description.abstractAFRIKAANS ABSTRACT: Cavalieri integrasie in R n bied ’n nuwe visualiseringsmeganisme vir geweegde integrasie en daag die idee van streng reghoekige integrasiestroke uit. Dit doen dit deur dieintegreerder binne die grenskrommes van die integraal te stoor. Hierdie artikel ondersoek die Cavalieri integraal as ’n superstel van Riemann integrasie in R n−1 , waar die integraal gedefinieer word deur ’n translasiegebied in R n−1 , wat die funksie wat geintegreer word, die integreerder en die integrasiestreek uniek definieer. In R 2 maak hierdie verfynde translasiestreekdefinisie voorsiening vir die visualisering van Riemann-Stieltjes integrale asook ander vorme van geweegde integrasie, soos die Riemann-Liouville fraksionele integraal en die wiskundige konvolusie. Programmatiese implementering van sulke visualiserings en berekeninge van integrale waardes word ook ondersoek en maak staat op kennis van numeriese integrasie metodes, algoritmiese differensiasie en numeriese wortelbevindings algoritmes. Vir die R 3 geval vereis sulke visualiserings oor veelhoekige streke ’n meganisme vir die triangulasie van ’n stel geneste veelhoeke. Dit vereis ook transformasies wat die gebruik van herhaalde integrasie moontlik maak vir die berekening van die integrasiewaarde oor die geproduseerde driehoekige streke. Hierdie verseker dat standaard 1-dimensionele integrasie roetines gebruik kan word om die integrasiewaarde oor ’n driehoek te bereken.en_ZA
dc.description.versionxiv, 149 pages : illustrationsen_ZA
dc.identifier.urihttp://hdl.handle.net/10019.1/127023en_ZA
dc.language.isoen_ZAen_ZA
dc.language.isoen_ZAen_ZA
dc.subject.lcshRiemann integralen_ZA
dc.subject.lcshTriangulation en_ZA
dc.subject.lcshConvolutions (Mathematics)en_ZA
dc.subject.lcshAutomatic differentiationen_ZA
dc.subject.lcshCalculus, Integralen_ZA
dc.titleImplementation of the Cavalieri Integralen_ZA
dc.typeThesisen_ZA
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