OPTIMALIZATION OF TRIANGULAR
IRREGULAR

NETWORKS FOR MODELING OF GEOMETRICAL
STRUCTURE OF GEORELIEF

Richard Feciskanin

The main aim of paper is
to show that triangular irregular network has to fulfil condition of correct
configuration as much as possible to be more accurate model of georelief (or
other surface) used for visualization and like a base for further modelling.

The idea of condition of
correct triangle configuration is based on fact that for exact representation
of part of georelief surface (including first partial derivatives, slope and
aspect) suppose that normal vector of triangle plain and real normal vector of
modelled surface in place of triangle centroid are the same.

So,
the main consequence of not fulfilling the condition of correct configuration
of triangle is that normal vector computed based on triangle plane is other
than real normal vector of georelief surface in that place. The normal vector
is very important parameter for visualization especially for computing
illumination. Therefore an offset of normal vectors produce none accurate model
of georelief surface for visualization and also for processes modelling.

We
want to present properties of condition of correct configuration of triangle
depending on modelled surface, ways to fulfilling this condition. We want to
present experiment for minimizing offset of normal vectors of triangular
irregular network created by Delaunay triangulation by Lawson local optimalization
which created more accurate model of georelief surface for visualization and
other purposes.

**Abstract**

One of the most
important advantages of triangular irregular networks (TIN) is the possibility
to adjust its shape to the shape of georelief (terrain) surface. Therefore TIN
is preferred in some kinds of digital terrain model. In complex digital model
of georelief we want to model not only its basic shape (elevation) but also its
geometrical structure (with the first and the second partial derivatives and
from them derived morphometric variables – slope, aspect and curvatures).

Therefore
for exact modelling the triangles have to fulfil some conditions: the condition
of representativeness, the basic condition for location and density of input
point field – triangle vertices, and also the condition of correct
configuration for the most accurate modelling of partial derivatives and
morphometric variables.

We
present the condition of correct configuration that is based on relation between
georelief surface and plane of triangle. Simplifying of georelief surface in
small neighbourhood of triangle centroid by osculating paraboloid we get simple
geometrical relation between georelief surface and plane of triangle.

We can use this in (data-depend)
triangulation; we show the way for identifying of positional shift which
creates numerical error of the first partial derivatives, slope and aspect, so
we can choose the most accurate triangle of possible triangles.

If there is dense grid
as input, we can choose almost arbitrary point for constructed triangles. We present
algorithm for geometrically ordaining of position of searched the third vertex
for two known vertices, to create triangle fulfilling the condition of correct
configuration.

The disadvantage of the
presented methods is need to know the shape of georelief surface to represent,
but that is just it for exact representation. The advantage is direct
calculation of position of searching point, we do not need searching algorithm.
Created TIN is well suited for exact modelling of geometrical structure of
georelief, so it is assumption it will be used in some type of models.