Civil Engineering Reference
In-Depth Information
S
2
r
3
r
S
S
3
S
1
Figure 5.87
Updating sphere S on plane S
1
S
2
S
3
.
The angle θ that S has to turn through just touching sphere S3 on the surface has to be
determined by an iterative process. Such a θ exists as there is no intersection at the initial
position where θ = 0, and there is intersection when S is rotated through an angle α. In fact,
θ lies between 0 and α. A simple iterative procedure can be devised as follows.
e = ||SS
3
|| − (r + r
3
)
(5.5)
When θ = 0, e > 0, and when θ = α, e < 0; hence, θ can be computed by linear interpola-
tion. With a new value for θ, say θ′, the position of S and its radius r are updated on the new
θ′ plane, and a new error e′ can be obtained from Equation 5.5. Further iteration by means
of the secant method using this new pair of values (θ′, e′) can be conducted until error e is
within the tolerance of some specified value. In the present implementation, a tolerance of
5% has been adopted. However, tighter control, say, 1%, can be applied if a stricter compli-
ance of the specified size must be observed.
More rotations about new axes S
1
S
3
or S
2
S
3
are performed until no further descent is pos-
sible as shown in Figure 5.88. From the worked examples shown in Section 5.7.4, a sphere can
rotate as many as nine times before finding its parking place. However, on average, a sphere
only takes two rotations to find the insertion site. In the descending process, colliding with
non-neighbouring spheres is possible and could happen. However, checking for this to happen
would not be done during the process of descending by rotation for efficiency consideration.
Instead, in the phase of Delaunay point insertion, all connections will be verified to make sure
that edges are of size compatible to the node spacing function. Should there be any discrepancy
detected, the descent by rotation has to be done again with additional care to avoid intersection.
Edges on the surface
S
1
S
2
S
4
S
3
Figure 5.88
Descent by rotation between planes.
