Page 41 - Dynamic Vision for Perception and Control of Motion

2.1 Three-dimensional (3-D) Space and Time 25

2.1.1 Homogeneous Coordinate Transformations in 3-D Space

Instead of the Cartesian vector r C = (x, y, z), the homogeneous vector
p
˜
˜
r ( p x , p y , p z , ) (2.1)
˜
h
is used with p as a scaling parameter. The specification of a point in one coordinate
system can be “transformed” into a description in a second coordinate system by
three translations along the axes and three rotations around reference axes, some of
which may not belong to any of the two (initial and final) coordinate systems.
2.1.1.1 Translations
This allows writing translations along all three axes by the amount 'r = ('x, 'y,
'z) in the form of a matrix · vector multiplication with the homogeneous transfor-
mation matrix (HTM) for translation:
§ 100 ' · x
¨ 01 0 ' y ¸
r ¨ ¸ . r ˜
1 0
¨ 001 ' z ¸ (2.2)
¨ ¸
© 00 0 1 ¹
The three translation components shift the reference point for the rotated origi-
nal coordinate system.

2.1.1.2 Rotations
Rotations around all axes may be described with the shorthand notation c =
cos(angle) and s = sin(angle) by the corresponding HTMs:
§ 1 0 0 0· c § 0 s 0· § c s 0 0·
¨ 0 c s 0 ¸ ¨ 0 1 0 0 ¸ ¨ s c 0 0 ¸
R ¨ ¸ ; R ¨ ¸ ; R ¨ ¸ .
¨ 0 sc 0¸ s ¨ 0 c 0¸ ¨ 0 0 1 0¸
x y z
¨ ¸ ¨ ¸ ¨ ¸ (2.3)
© 0 0 01 ¹ © 00 0 1 ¹ © 0 0 0 1 ¹
The position of the 1 on the main diagonal indicates the axis around which the
rotation takes place.
The sequence of the rotations is of importance in 3-D space because the final re-
sult depends on it. Because of the dominant importance of gravity on Earth, the
usual nomenclature for Euler angles (internationally standardized in mechanical
engineering disciplines) requires the first rotation be around the gravity vector, de-
fined as “heading angle” \ (or pan angle for cameras). This reference system is
dubbed the “geodetic coordinate system”; the x- and y-axes then are in the horizon-
tal plane. The x-direction of this coordinate system (CS) may be selected as the
main direction of motion or as the reference direction on a global scale (e.g., mag-
netic North). The magnitude of rotation \ is selected such that the x-axis of the ro-
tated system comes to lie vertically underneath the x-axis of the new CS [e.g., the
body-fixed x-axis of the vehicle (x O in Figure 2.3, upper right corner)]. As the sec-
ond rotation, the turn angle of the vehicle’s x-axis perpendicular to the horizontal
plane has proven to be convenient. It is called “pitch angle” T for vehicles (or tilt