Graphics Programs Reference
In-Depth Information
A convenientformula of computing
A
from the corner coordinates (
x
i
,
y
i
) is
111
x
1
1
2
A
=
x
2
x
3
(6.49)
y
1
y
2
y
3
The area coordinates are mappedinto the Cartesian coordinates by
3
3
x
(
α
,α
,α
3
)
=
1
α
i
x
i
y
(
α
,α
,α
3
)
=
1
α
i
y
i
(6.50)
1
2
1
2
i
=
i
=
The integration formula over the element is
A
k
f
[
x
(
α
)
,
y
(
α
)
]
dA
=
W
k
f
[
x
(
α
k
)
,
y
(
α
k
)
]
(6.51)
A
where
α
k
represents the area coordinates of the integrationpoint
k
, and
W
k
are the
weights. The locationsoftheintegrationpoints are shown in Fig. 6.10, and the corre-
sponding values of
α
k
and
W
k
arelistedinTable 6.7. The quadrature in Eq. (6.51) is
exact if
f
(
x
,
y
) is apolynomialofthedegree indicated.
b
a
c
a
a
Figure 6.10.
Integrationpoints of trian-
gular elements.
c
d
b
(a) Linear
(b) Quadratic
(c) Cubic
Degree of
f
(
x
,
y
)
Point
α
k
W
k
(a) Linear
a
1
/
3
,
1
/
3
,
1
/
3
1
(b)Quadratic
a
1
/
2
,
0
,
1
/
2
1
/
3
b
1
/
2
,
1
/
2
,
0
1
/
3
c
0
,
1
/
2
,
1
/
2
1
/
3
/
,
/
,
/
−
/
(c) Cubic
a
1
3
1
3
1
3
27
48
b
1
/
5
,
1
/
5
,
3
/
5
25
/
48
c
3
/
5
.
1
/
5
,
1
/
5
25
/
48
d
1
/
5
,
3
/
5
,
1
/
5
25
/
48
Table 6.7
triangleQuad
The function
triangleQuad
computes
A
,
y
)
dx dy
overatriangular regionus-
ing the cubicformula—case (c) in Fig. 6.10. The triangle is definedbyits corner co-
ordinate arrays
x
and
y
, where the coordinates must belistedinacounterclockwise
directionaround the triangle.
f
(
x
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