Analysis of flow and visibility on triangulated terrains

Landscapes and their morphology have been widely studied for predicting physical phenomena, such as floods or erosion, but also for planning human activities effectively, such as building prominent fortifications and watchtowers. Nowadays, the study of terrains is done in a computer-based environment; terrains are modelled by digital representations, and algorithms are used to simulate physical processes like water flow and to compute attributes like visibility from certain locations. In the current thesis we focus on designing new algorithms for computing structures related to water ow and visibility on digital terrain representations. Most specifically, the terrain representations that we considered are the so-called Triangulated Irregular Networks (tins), that is, piecewise linear surfaces that consist of triangles. One of the problems that are considered is the effect of noise on the worst-case complexity of visibility structures on tins. The view that a person can have from a point on the surface of a tin can be very complex, since in the worst case thin obstacles in the foreground may appear to fragment many long terrain edges in the background into visible and invisible pieces. In our analysis we considered tins whose triangles have some well-defined properties that terrains in practice are expected to have. Although complex visibility structures can be induced on such tins as well, we proved formally that slight perturbations on the elevations of the tin vertex set will always get rid of the high complexity. Another key problem that is studied is to design efficient algorithms that compute flow-related structures on tins. So far it was known that, in the case of tins, drainage structures that were computed using a consistent flow-model could have high complexity for specific input instances. We managed to develop a mechanism that can extract important information on flow paths and other drainage structures without computing those structures explicitly. This mechanism can be used as a basis for designing a variety of efficient algorithms, such as for computing the area measure of drainage structures or for computing structures that represent the terrain topology. The last part of the presented work involves the implementation of a software package that computes drainage structures on tins. In this package flow is modelled as following strictly the direction of steepest descent on the tin surface. Existing software for related applications either constrain flow on the edge set of the tin, or use inexact arithmetic, both of which introduces imprecise and/or incorrect results in the output. Our implementation is the first one that, at the same time, follows a robust flow model and uses exact arithmetic. We have used this implementation as a point of reference for evaluating experimentally the quality of the output of other flow models which are used in many hydrological applications. We have also used our software for conducting experiments on extracting watersheds on imprecise tins, that is, tins where the elevation values of the vertices are not exactly defined but are subject to noise from some given interval. Based on the results of these experiments, we have designed a novel method for extracting watersheds on imprecise terrains that produces high quality output.