Full Text Available
Note: Clicking the button above will open the full text document at the original institutional repository in a new window.
Implementation of the Cavalieri Integral
Thesis (MSc)--Stellenbosch University, 2023.
Saved in:
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Thesis |
| Language: | en_ZA en_ZA |
| Published: |
2023
|
| Subjects: | |
| Tags: |
No Tags, Be the first to tag this record!
|
| _version_ | 1867614015317868544 |
|---|---|
| access_status_str | Open Access |
| author | van Zyl, Christoff |
| author2 | Grobler, Trienko |
| author_browse | Grobler, Trienko van Zyl, Christoff |
| author_facet | Grobler, Trienko van Zyl, Christoff |
| author_sort | van Zyl, Christoff |
| collection | Thesis |
| description | Thesis (MSc)--Stellenbosch University, 2023. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/127023 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA en_ZA |
| last_indexed | 2026-06-10T12:45:19.124Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2023 |
| publishDateRange | 2023 |
| publishDateSort | 2023 |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/127023 Implementation of the Cavalieri Integral van Zyl, Christoff Grobler, Trienko Stellenbosch University. Faculty of Science. Dept. of Computer Science. Riemann integral Triangulation Convolutions (Mathematics) Automatic differentiation Calculus, Integral Thesis (MSc)--Stellenbosch University, 2023. ENGLISH 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. AFRIKAANS 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. xiv, 149 pages : illustrations 2023-02-19T18:12:08Z 2023-05-18T07:00:35Z 2023-02-19T18:12:08Z 2023-05-18T07:00:35Z 2023-02 Thesis http://hdl.handle.net/10019.1/127023 en_ZA en_ZA application/pdf |
| spellingShingle | Riemann integral Triangulation Convolutions (Mathematics) Automatic differentiation Calculus, Integral van Zyl, Christoff Implementation of the Cavalieri Integral |
| title | Implementation of the Cavalieri Integral |
| title_full | Implementation of the Cavalieri Integral |
| title_fullStr | Implementation of the Cavalieri Integral |
| title_full_unstemmed | Implementation of the Cavalieri Integral |
| title_short | Implementation of the Cavalieri Integral |
| title_sort | implementation of the cavalieri integral |
| topic | Riemann integral Triangulation Convolutions (Mathematics) Automatic differentiation Calculus, Integral |
| url | http://hdl.handle.net/10019.1/127023 |
| work_keys_str_mv | AT vanzylchristoff implementationofthecavalieriintegral |