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approximate the surface of the terrain being modelled. We distinguish between 2D
and 3D triangulation. In 2D triangulation, the area covering the input data set in the
XoY plane is divided into non-overlapping triangles. In other words, the triangulation of a set of points is a system of triangles whose vertices form a corresponding
set, i.e. whose interiors do not intersect with each other and whose union completely
covers the surface of the terrain. At the same time, using the values of heights at
given points of the grid, spatial triangles are obtained that approximate the surface
of the terrain (Li et al. 2005).
Also, in order to better approximate the relief, the terrain model is represented by
triangular surface patches. This means, in addition to the height of the vertices of the
TIN triangles, additional conditions relate to the values of the normal to the terrain
surface at the TIN nodes and the conditions that require minimization of the curvature of the terrain surface. There are a number of criteria and algorithms for forming
2D TIN. The 2D Delaunay triangulation, which has particularly interesting geometric properties, is most commonly formed. Namely, Delaunay’s triangulation maximizes the minimum angle of the triangles of the grid, i.e. it eliminates “elongated”
triangles of the network. Also, in practice, for modelling the terrain, modifications
of Delaunay’s triangulation are often used in order to achieve some specific requirements in order to obtain the most accurate representation of the terrain surface. In
some cases, it is required to install mandatory lines and polygons (structural characteristics of the terrain) in the model itself, so that these lines must be represented by
the sides of the triangle TIN (Li et al. 2005).
When it comes to 3D triangulation, it consists of spatial triangles whose
projections in the XoY plane can generally intersect, i.e. overlap. As in the case of
2D triangulation, Delaunay’s tetrahedralization or 3D Delaunay’s triangulation is
most often used here. The data structure that represents the TIN can be based on
storing data on the edges of triangles (triangle edges) or on storing data on the
triangles themselves. In other words, in addition to the table containing the points
(nodes) of the TIN, it is necessary to keep the table with the sides of the triangles for
the first approach or the table with the triangles of the TIN for the second approach.
In both cases, information that is not stored directly (data on triangles, i.e. on the
sides of the network) can be obtained based on the network topology. In order to
preserve information related to points, lines and surfaces that may be important for
modelling the terrain surface, for TIN elements (nodes, sides and triangles), it is
necessary, in addition to their geometry and topology, to keep the appropriate
attributes, i.e. heights and mutual relations of the mentioned TIN elements (Li
et al. 2005).
6.6.2 Interpolation Methods and Their Basic Features
The term interpolation comes from the Latin word “inter” which means between
and the Greek word “polos” which refers to a point or a node. In other words, interpolation is defined as the process of determining a new (unknown) value between
6 Evidence for Atmospheric Depositions Using Attic Dust, Spatial Mapping…
approximate the surface of the terrain being modelled. We distinguish between 2D
and 3D triangulation. In 2D triangulation, the area covering the input data set in the
XoY plane is divided into non-overlapping triangles. In other words, the triangulation of a set of points is a system of triangles whose vertices form a corresponding
set, i.e. whose interiors do not intersect with each other and whose union completely
covers the surface of the terrain. At the same time, using the values of heights at
given points of the grid, spatial triangles are obtained that approximate the surface
of the terrain (Li et al. 2005).
Also, in order to better approximate the relief, the terrain model is represented by
triangular surface patches. This means, in addition to the height of the vertices of the
TIN triangles, additional conditions relate to the values of the normal to the terrain
surface at the TIN nodes and the conditions that require minimization of the curvature of the terrain surface. There are a number of criteria and algorithms for forming
2D TIN. The 2D Delaunay triangulation, which has particularly interesting geometric properties, is most commonly formed. Namely, Delaunay’s triangulation maximizes the minimum angle of the triangles of the grid, i.e. it eliminates “elongated”
triangles of the network. Also, in practice, for modelling the terrain, modifications
of Delaunay’s triangulation are often used in order to achieve some specific requirements in order to obtain the most accurate representation of the terrain surface. In
some cases, it is required to install mandatory lines and polygons (structural characteristics of the terrain) in the model itself, so that these lines must be represented by
the sides of the triangle TIN (Li et al. 2005).
When it comes to 3D triangulation, it consists of spatial triangles whose
projections in the XoY plane can generally intersect, i.e. overlap. As in the case of
2D triangulation, Delaunay’s tetrahedralization or 3D Delaunay’s triangulation is
most often used here. The data structure that represents the TIN can be based on
storing data on the edges of triangles (triangle edges) or on storing data on the
triangles themselves. In other words, in addition to the table containing the points
(nodes) of the TIN, it is necessary to keep the table with the sides of the triangles for
the first approach or the table with the triangles of the TIN for the second approach.
In both cases, information that is not stored directly (data on triangles, i.e. on the
sides of the network) can be obtained based on the network topology. In order to
preserve information related to points, lines and surfaces that may be important for
modelling the terrain surface, for TIN elements (nodes, sides and triangles), it is
necessary, in addition to their geometry and topology, to keep the appropriate
attributes, i.e. heights and mutual relations of the mentioned TIN elements (Li
et al. 2005).
6.6.2 Interpolation Methods and Their Basic Features
The term interpolation comes from the Latin word “inter” which means between
and the Greek word “polos” which refers to a point or a node. In other words, interpolation is defined as the process of determining a new (unknown) value between
6 Evidence for Atmospheric Depositions Using Attic Dust, Spatial Mapping…