Appendix A
TRIGONOMETRIC CALCULATIONS
FOR URBAN LINES OF SIGHT
This appendix explains the trigonometric calculations behind much
of the information presented in Chapter Four.
Figure A.1 gives an example of how to determine the maximum horizontal distance, line AD, a UAV can be away from a street and still see
three-fourths of that street over a building of a given height.
RAND MR1187-A.1
E
Know:
Line AB = 12.5 m
Line BC = 20 m
Line DE = 1,000 m
Angles b and d are right angles
e
C
c
Building
3/4 of street
A
a b
B
d
D
Figure A.1—Determining Maximum Horizontal Distance
for Viewing Streets over a Building
207
208
Aerospace Operations in Urban Environments
The figure uses the average building height of 20 m and street width
of 50 m given in Table 4.2 for UTZ V, to illustrate the computation. To
be able to see three-fourths of the street means that line AB is onefourth of 50 m, or 12.5 m, that will not be seen. Line BC is the average
building height, 20 m. The UAV altitude, defined as line DE, is 1,000
m, and angles b and d are right angles. With this information, we can
use simple trigonometry to find the length of line AD, as follows:
First we find the tangent of angle a. The tangent of angle a is the
height of triangle ABC divided by its base. In this case, this is the
20-m building height divided by one-fourth of the street width,
12.5 m.
Since angle a is the same in triangles ABC and ADE, we know that the
tangent of angle a must also equal the UAV altitude (DE) divided by
the maximum horizontal standoff (AD), or 1.6:
tan a =
BC 20 m
=
= 1.6
AB 12.5 m
(A.1)
Simple algebra shows that AD equals 625 m:
tan a = 1.6 =
DE 1, 000 m
=
AD
AD
(A.2)
1, 000 m
AD
(A.3)
1, 000 m
= AD
1.6
(A.4)
AD = 625 m
(A.5)
1.6 =
The arctangent function on any scientific calculator (or table) can be
used to find that angle a is about 58˚. This is the minimum angle at
which the UAV can see three-fourths of the street. If the UAV moves
closer than 625 m, it will be able to see more of the street; as it moves
farther away, it will be able to see progressively less of the street.
Similar calculations form the basis of the information presented
throughout Chapter Four.