Surfaces - Topography and topology

Abstract

ISO 25178-2 specifies areal field parameters as well as areal feature parameters. While for the first group the whole set of points defining a scale-limited surface is considered, for the second group only a subset of surface features is taken into account. As a consequence, an adequate data structure for surface characterisation in combination with an appropriate method for surface simplification are required. Three data structures for surface characterisation, namely, Morse-Smale complexes, weighted surface networks and change trees are discussed. Hereafter, the focus is laid on approaches for determining the relevance of topological features with respect to surface topography. Another topic of interest is surface simplification, i.e. the process of deriving from an original surface a second surface of decreased complexity, but with its structural properties being retained. Within the geosciences this concept is associated with the transition from large-scale maps to small-scale maps, whereas in the technical sciences it corresponds to the reduction of measurement noise. From a topological point of view, a theorem proven by Matsumoto may be regarded as the core theorem with respect to surface simplification. Its combination with the two concepts of relevance of a topological feature and degree of simplicity represents the basis of a formal procedure for surface simplification as required in ISO 25178-2 and ISO 16610-85.