| Popis: |
V tem diplomskem delu se ukvarjamo s sledenjem žarku v neevklidskih prostorih. Sledenje žarku je metoda, ki simulira potovanje svetlobnih žarkov in se uporablja v računalniški grafiki za izris realističnih slik. Ponavadi je implementirana v običajnem evklidskem prostoru. Spoznamo geodetke, ki omogočajo sledenje žarku v neevklidskih prostorih. Izpeljemo splošen sistem diferencialnih enačb za njihov izračun in predstavimo numerične metode za reševanje tega sistema. Algoritem implementiramo in uporabimo za evklidski prostor, ploščati torus in dvodimenzionalno sfero ter razložimo dobljene rezultate. Vizualno predstavimo lastnosti prostorov, ki so sicer težje razumljive le iz pripadajočih matematičnih modelov. In this diploma thesis we delve into ray tracing in non-Euclidean spaces. Ray tracing is a method that simulates traveling of light rays and is used in computer graphics to draw realistic images. Usually it is implemented in standard Euclidean space. We present the notion of geodesic curves, which allow us to trace rays in non-Euclidean spaces. Then, a general system of differential equations determining geodesics is derived and numerical methods for solving it are presented. We implement and apply the algorithm to the Euclidean space, a flat torus and a two-dimensional sphere, and then explain the results. Finally, we visually present the properties of these spaces, which are more difficult to understand using only the corresponding mathematical models. |
