Graphics Reference
In-Depth Information
2
4
3
5 ¼
2
4
3
5
2
4
3
5
0
cos y
0
sin y
0
x
y
1
x
0
0
1
0
0
y
(2.14)
0
sin y
0
cos y
0
z
0
0
0
1
1
2
4
3
5 ¼
2
4
3
5
2
4
3
5
0
x
cos y
sin y 00
x
y
1
0
y
sin y
cos y 00
(2.15)
0
z
0
0
1
0
1
0
0
0
1
Combinations of rotations and translations are usually referred to as rigid transformations because dis-
tance is preserved and the spatial extent of the object does not change; only its position and orientation
in space are changed. Similarity transformations also allow uniform scale in addition to rotation and
translation. These transformations preserve the object's intrinsic properties 2 (e.g., dihedral angles 3 )
and relative distances but not absolute distances. Nonuniform scale, however, is usually not considered
a similarity transformation because object properties such as dihedral angles are changed. A shear
transformation is a combination of rotation and nonuniform scale and creates columns (rows) that
might not be orthogonal to each other. Any combination of rotations, translations, and (uniform or non-
uniform) scales still retains the last row of three zeros followed by a one. Notice that any affine trans-
formation can be represented by a multiplicative 3
3 matrix (representing rotations, scales, and
shears) followed by an additive three-element vector (translation).
2.1.5 Representing an arbitrary orientation
Rigid transformations (consisting of only rotations and translations) are very useful for moving objects
around a scene without disturbing their geometry. These rigid transformations can be represented by a
(possibly compound) rotation followed by a translation. The rotation transformation represents the
object's orientation relative to its definition in object space. This section considers a particular way
to represent an object's orientation.
Fixed-angle representation
One way to represent an orientation is as a series of rotations around the principal axes (the fixed-angle
representation ). When illustrating the relationship between orientation and a fixed order of rotations
around the principal axes, consider the problem of determining the transformations that would produce
a given geometric configuration. For example, consider that an aircraft is originally defined at the ori-
gin of a right-handed coordinate system with its nose pointed down the z -axis and its up vector in the
positive y -axis direction (i.e., its object space representation). Now, imagine that the objective is to
position the aircraft in world space so that its center is at (20,
10, 35), its nose is oriented toward
the point (23,
14, 40), and its up vector is pointed in the general direction of the y -axis (or, mathe-
matically, so that its up vector lies in the plane defined by the aircraft's center, the point the plane is
oriented toward, and the global y -axis) (see Figure 2.6 ) .
2 An object's intrinsic properties are those that are measured irrespective of an external coordinate system.
3 The dihedral angle is the interior angle between adjacent polygons measured at the common edge.
 
Search WWH ::




Custom Search