Civil Engineering Reference
In-Depth Information
where vector cross product AB × AC represents twice the signed area, and
23
is a normalis-
ing factor such that an equilateral triangle on a planar domain will have a maximum value
of 1.
Borouchaki et al. (2000a) tried to generalise the α-quality by evaluating the signed area
of the triangle and the length of the edges with respect to the given metric field, and the
α-quality for triangle ABC under metric M is given by
AB
×
AC
M
α=
23
2
2
2
AB
+
BC
+
CA
M
M
M
Since M varies over the triangular element, Borouchaki et al. (2000a) further proposed
that the α-quality of triangle ABC be sampled only at the vertices A, B and C such that
AB
×
AC
MA
()
α
A
=
23
2
2
2
AB
+
BC
+
CA
MA
()
MA
()
MA
()
where α
A
is the α-quality, and M(A) is the metric at point A. Similarly, α-qualities of the
triangle are evaluated at points B and C, and a minimum norm for the α-quality of triangle
ABC for metric M is given by α
= min(α
A
, α
B
, α
C
).
However, there is an obvious weakness in this definition. Consider triangles ABC and
ABD, as shown in Figure 4.21, in which the ellipse represents the locus of unit length from
the centre O for constant metric field M. With reference to Section 4.2.7, triangles ABC and
ABD are of the same area λμ for metric M with principal stretches λ and μ, and as for edges
‖AC‖
M
= ‖BD‖
M
= 2, ‖AD‖
M
= ‖BC‖
M
; hence, they are of the same α-quality. Nevertheless,
triangle ABD is preferred over triangle ABC in terms of shape quality as it possesses larger
internal angles.
Shape quality is something that can easily be appreciated and judged in the Euclidean
space, and a quality measure of triangles with respect to a general metric may not be that
appropriate and be well-defined with a solid mathematical basis. Therefore, instead of defin-
ing a shape measure for triangles generated over a parametric domain with anisotropic
metric, the shape of a triangle is assessed by the actual transformed geometry in the 3D
Euclidean space, as shown in Figure 4.22. Hence, the shape measure of a triangle on a para-
metric domain is given by
φφ φφ
φφ φφ φφ
23
2
()()
AB
×
()()
AC
23
AB
×
AC
α
=
=
2
)
2
2
2
2
()()
AB
+
()()
BC
+
()(
C
A
AB
+
BC
+
CA
B
µ
λ
C
A
O
D
Figure 4.21
α
-Quality of triangles in metric field.
