Civil Engineering Reference
In-Depth Information
Given point A and unit vector
u
, point B can be determined by iteration:
1
T
i. Get unit length s at A in the direction
u
,
()
sMs
u
()
u
=
1
s
=
A
T
uu
M
ii. Compute point B = A + s
u
A
iii. Evaluate the distance between points A and B, r = ‖AB‖
M
iv. Update s ↦ 2s/(1 + r)
v. If
1−r
, go to step ii
In general, there is a residue between point P
n+1
and the ending point of the line seg-
ment E. However, this residue cannot be distributed to the points by direct relocation
similar to the case of isotropic metric as the distance measure is position-dependent for
general metric field. Instead, we have to compute the discrepancy for each point given by
PE
n
n
1
δ=
M
The points have to be regenerated along the line segment following iteration steps i
to v with adjusted distance 1 + δ or 1 − δ if P
n+1
goes beyond E, and the last point P
n+1
should coincide with ending point E.
3.
Triangulation of the boundary points.
The initial triangulation of the boundary points
can be constructed by means of the classical DT using Euclidean metric, as shown in
Figure 4.32.
4.
Insertion of interior points.
Generation of interior points follows the procedure of
the Delaunay-ADF method, as elaborated in Section 3.7.3, in such a way that the
underlying metric is respected. Given a frontal (boundary) line segment AB, which is
of more or less unit length, a point C is created, which is of unit length from points A
and B. For anisotropic metric field, the distance measure is non-linear, and point C, in
general, cannot be determined analytically; hence, an iterative procedure is adopted in
locating point C.
The initial position of point C can be estimated by means of a simple Euclidean
norm such that
‖AC‖ = ‖AB‖ and ‖BC‖ = ‖AB‖
A more sophisticated procedure for locating the initial position of point C is by
means of the intersection of the unit metric at points A and B. Ellipses of the unit
Figure 4.32
Triangulation of boundary points.
