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FIGURE 6.30
Triangle used to define coordinates.
are often referred to as
triangular
or
homogeneous coordinates
[Schwarz, 1980]. These are very
useful in establishing shape functions referred to nodal DOF.
Consider the triangle of Fig. 6.30. We wish to locate a point within the triangle. To do
so, draw lines from a given point to the three vertices, thereby dividing the triangle into
three triangles of areas
A
1
,A
2
, and
A
3
. The sides of the triangles are designated by the same
number as the opposite vertex. The areas are identified by the number of the adjacent side.
The quantities
L
i
,i
=
1
,
2
,
3
,
L
1
=
A
1
/
A,
L
2
=
A
2
/
A,
L
3
=
A
3
/
A
(6.72)
where
A
is the area of the original triangle, are defined to be the triangular coordinates.
These two-dimensional natural coordinates are similar to the one-dimensional coordinates.
Note that
A
1
+
A
2
+
A
3
=
A
so that
L
1
+
L
2
+
L
3
=
1
(6.73)
The relationships between the Cartesian coordinates
x, y
, which are the coordinates of
the points in the element, and the triangular coordinates
L
1
,L
2
, and
L
3
are
x
=
L
1
x
1
+
L
2
x
2
+
L
3
x
3
(6.74)
=
+
+
y
L
1
y
1
L
2
y
2
L
3
y
3
where
x
i
,y
i
,i
1
,
2
,
3 are the coordinates of the nodes. These can be verified at the vertices,
e.g., at corner, 1
,A
2
=
=
A
3
=
0
,A
1
=
A
so that
L
2
=
L
3
=
0
,L
1
=
1. Thus,
x
=
x
1
,asit
should.
The triangular coordinates can be expressed in terms of the known locations of the ver-
tices. From Eqs. (6.73) and (6.74),
1
x
y
111
x
1
L
1
L
2
L
3
=
x
2
x
3
y
1
y
2
y
3
or, by inversion,
1
2
A
(α
=
+
β
+
γ
)
=
L
i
i
x
i
y
i
1
,
2
,
3
(6.75)
i
with
111
x
1
1
2
(
1
2
A
=
x
2
y
3
+
x
3
y
1
+
x
1
y
2
−
x
2
y
1
−
x
3
y
2
−
x
1
y
3
)
=
x
2
x
3
y
1
y
2
y
3
x
2
x
3
11
y
2
11
x
2
α
1
=
β
1
=−
γ
1
=
y
2
y
3
y
3
x
3
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