Civil Engineering Reference
In-Depth Information
If edge AB is of unit length with respect to M, i.e. L = 1, we have ρ ≥ ‖AB‖. The metric
M allows us to control the edges of a mesh on S such that the unit length in any direction is
always smaller than the radius of curvature at the point under consideration.
4.2.5 Geometrical control
Two common measures can be applied to control the geometrical closeness to a curved sur-
face (Miranda et al. 2009). The first criterion is to specify the gap between an edge and the
curved surface, as shown in Figure 4.10. Let h be the length of the edge and ρ be the radius
of curvature in the direction of the edge. We have
2
2
h
h
2
2
2
d
=− =−
ρ
d
ρ
2
4
2
2
2
h
α
δ
ρ
α
2
δρ ρρ ρ
=−=− −=−−
d
11
4
or
=− −≤ =
11
4
εα
h/
ρ
4
2
2
α
α
εε αε
2
1
−≤ −
ε
1
44
2
≤
−
≤
2
(
2
−
ε
)
h
≤
2
ε
(
2
−
ερ
)
Hence, in order that the gap δ between a line segment and the surface be bounded by the
ratio ε = δ/ρ, h = αρ could not exceed the value given by
αε
=
2(
−
ε
)
For example, if ε is set at 1%, α = 0.282, i.e. h cannot exceed 0.282ρ; otherwise, the gap
between the line segment and the curved surface will be more than 1% of ρ.
Alternatively, the ratio of the length of a chord to that of the corresponding curve can be
specified, as shown in Figure 4.11. Let h be the length of an edge (chord) and s be that of the
shortest curve on the surface connected to the same end points of the edge. As the curve is
always longer than the chord, we can control the length of an edge by specifying a value ε
such that
sh
s
−
≤
ε
or
h
≥ −
(
1
ε
)
s
h
δ
d
ρ
Figure 4.10
Gap between an edge and a curved surface.
