Civil Engineering Reference
In-Depth Information
overlapping to existing spheres. Imagine that the surface of the cluster is bent and spread onto a
flat bed, which is same as the process of developing the earth surface into maps; the problem is
similar to tossing a ball into a pool of balls of various sizes and then finding out where the tossed
ball will come to land. The agent that set up the motion is the gravitational force, which pulls
the ball to the lowest possible position, and the geometry permitting this to happen is the path
between existing spheres about which the tossed ball rotates around and descends on the way
to the lowest point. Following the same principle for the packing of spheres, the newly inserted
sphere will move down the track by rotation starting from an edge on the front relatively close
to the origin. The inserted sphere will acquire a lower position each time it rotates between two
spheres, and it may change axis along its way until further descent is not possible where rotation
about any one of the three axes that block the sphere will raise it to a higher position.
5.7.2.5.3 Rotate between spheres
As shown in Figure 5.84, rotation between two spheres S
1
and S
2
with, respectively, radii r
1
and r
2
is possible provided that the new sphere S with radius r satisfies a simple condition:
r
1
+ r
2
+ 2r > d
12
where d
12
is the distance between spheres S
1
and S
2
(5.3)
Descent by rotation starts from the nearest edge from the front. Let S
1
be the frontal point
closest to the origin. Examine all the frontal nodes connected to S
1
and select S
2
, which is closest
to the origin. Then S
1
S
2
can serve as the initial rotation axis for the new sphere S. In case condi-
tion 5.3 cannot be satisfied, pick another sphere connected to S
1
to form the axis of rotation.
5.7.2.5.4 Rotation about an axis
Since we would like to pack spheres of different size specified by the node spacing function,
the radius of the newly inserted sphere has to be computed according to its current physical
position in space. As the sphere moves around in the rotation process, its radius has to be
updated from one axis to the next in order to be consistent with the size requirement. The
radius of the inserted sphere can be estimated by evaluating the desirable size at the mid-
point of S
1
and S
2
. The radius of S has to be updated as it moves away from the mid-point to
a new position. A couple of iterations may be necessary to set up the initial position with a
radius compatible with the node spacing function and tangent to spheres S
1
and S
2
as shown
in Figure 5.84. As the radius r of the proposed sphere S tangent to S
1
and S
2
is often needed,
the details for its determination are given as follows. Spheres S
1
, S
2
and S are lying on the
same plane, and hence, two simple conditions have to be satisfied.
||SS
1
|| = r + r
1
and ||SS
2
|| = r + r
2
(5.4)
S
S
r
r
(b)
(a)
r
1
S
r
2
S
1
S
2
d
12
S
1
S
2
S
1
S
2
Figure 5.84
Proposed sphere S touching existing spheres S
1
and S
2
: (a) S is away from S
1
and S
2
; (b) S is close
to S
1
and S
2
.
