Complexity of the Geological Framework and Use of Mathematics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 35

Fig. 1.21 Illustration of definitions of sides, height and top angles for three triangles taken
from artificial example of Fig. 1.20 . Event points at (a)C 1 ,(b)C 2 and (c)C 3 , respectively
(Source: Agterberg 1994 , Fig. 5)

Table 1.1 Formulae for ˉ ijk
if h ik < r ij max( a , b ) with
further conditions as given
in row and column headings

| ʸ ik | ¼α ik + ʲ | ʸ ik |

< α

ik + ʲ ik

a ik < r ik b ik ˉ ijk ¼ ( ʳ ijk + α ik ) s ik ˉ ijk ¼ ( ʳ ijk α ik ) s ik
b ik < r ik a ik ˉ ijk ¼ ( ʳ ijk + ʲ ik ) s ik ˉ ijk ¼ ( ʳ ijk ʲ ik ) s ik
h ij < r ij min( a , b ) ˉ ijk ¼ 2 ʳ ijk s ik ˉ ijk ¼ 0
Latter distinguish between triangles with acute angles only
( cf . Fig. 1.20a, c ) and triangles with acute angle along their base
( cf . Fig. 1.20b ); s ik is sign of

ik

ʸ ik (Source: Agterberg 1994 ,

Table 1)

points (see Fig. 1.21 ). According to elementary trigonometry, the shortest distance
( h ik ) between C i and he line through A k and B k satisfies:

p

s ik s ik

2
a ik b ik

a ik þ b ik þ c ik

h ik ¼ ð a ik Þ ð s ik b ik Þ ð s ik c ik Þ ; s ik ¼

2

The following acute angles can be defined for each triangle:

cos 1 h ik cos 1 h ik cos 1 h ik

r ik

α ik ¼

a ik ,

ʲ ik ¼

b ik ,

ʳ ik ¼

For a given triangle (with C i , A k and B k ) only those circles around C i with radius
r ij that intersect the side of the polygon at a point between A k and B k are used for the
calculation of the weight w ij . Let ˉ ijk represent the portion of ʸ ik contributing to the

reduction of w ij 1 (see Fig. 1.22 for examples). A set of rules is given in

Table 1.1 . The relative importance of these rules decreases when the sides of the

Σ k ˉ ijk /2 ˀ