of
on
rs.
id
ry
re
(3)
1a)
ib)
4c)
)r-
nd
ur
ng
cle
of
6)
From the definition of er follows, that the direction
vector d of the straight line will always lie in the
(ez,eg)-plane. Angle y, which is defined as the angle
between the eg§-axes and the direction vector d in the
e§-ep-plan is used as the fourth parameter. Figure 2
shows the final geometric relations.
The rotation matrix
sind cosó cos cosó -sind
Tsc - | sinósinó cosósinó cosó |, (5)
cosó -sinö 0
transforms spherical coordinates (S) into cartesian (C)
ones. Using the transformation matrix Tsc, direction
vector d can be express as a function of 6, à, y,
Xi 0
Ya! = T.C la * cosy (6)
la * siny
where la = 1 is the length of d. The point S and the
direction vector d are now expressed as functions of
the three spherical coordinates 8, $, r and the angle y.
Now it is possible to exchange the temporary line para-
meters ( A, B, C, D) by (6, 6, r, y) , i.e. equation (3) is
changed to
IS(8$,$,r)-C,d(6,0 y) ml = 0 (7)
Describing a straight line with the four parameters 6, à,
r, y results in the explicit analytical equations given in
equations (4) and (6).
2.3 Geometric constraints
Man-made objects are often characterized by straight
lines. They can be vertical, horizontal, parallel to each
other or they can intersect in a point. Therefore the in-
troduction of geometric constraints seems reasonable.
In a first group geometric conditions for one straight
linear feature and in the second group constraints con-
cerning two ore more linear features are presented.
First Group of constraints - Independent lines The
condition horizontal means, that the direction vector
d must be parallel to the X-Y-plan. The eg-axis is by
definition parallel to the X-Y-plan, therefore angle y
must be set to y = 50 gon or y= 150 gon.
In the vertical case the angle 5 have to be set to
ô = 100 gon. Therefore the es-axis is parallel to the Z-
axes. This leads to the second condition necessary to
achieve a vertical line, y= 0 gon or y = 200 gon.
The last constraint in this group is a chosen angle
between the direction vector d; of the searched line
671
feature and a known direction d. The chosen angle œ
between the two direction vectors can be calculated by
died
emo 3 deii €
If direction vector of the searched straight line has
index i and the fixed direction d has no index, d; can be
expressed as
ld;1 Id! (cosx-(a+b+c)) = 0 (9)
where
a = (cosd; cos; cosy; - sing; siny; ) X4
b = (cosd; sind; cosy; + cosd; siny; ) Ya
C= ( -sin$j cosy; ) Za
Idil = Id! = 1 length of the two vectors.
The constraints, parallel and perpendicular, are also
achieved in this case by setting the angle a = 0 gon and
c — 100 gon respectively.
The condition for the parallel case will look like
Idil Idi. (1 -(atb-c)) = 0 (10)
and for the perpendicular case
Idjl Id! (a-b«c) - 0 (11)
The help variables ( a, b, c) in (10) and (11) are identical
with them used in (9).
Second Group of constraints - dependent lines In
the first constraint the angle a between two straight
line features is chosen. Here two variable 3D straight
lines are used. The direction vector of the first line is
denoted to di, while the second line is denoted to d;,1-
New conditions are formulated based on (8). The help
variables a,b,c are changed to ef g.
Idil Idi4l (cosa -(e+f+g)) = 0 (12)
where
e = (CcosSöÖj+1 COS®j+1 COSYi+1 - Sin®j+1 Sinyi+1)
* ( cosó; cos$; cosy; - sin$; siny; )
f - (cos8j,, sinój,1 cosy,1 + COSOj,] SINY+] )
* (cosd; sing; cosy; + cos¢; siny; )
g = ( -sing; cosy; ) ( -Sin$;,1 COSYi+1 )
