Matrices and Linear Transformations - 3D Math Primer for Graphics and Game Development - page 158

angle-preserving matrix preserves proportions. We do not consider reflec-
tion an angle-preserving transformation because even though the magnitude
of angle between two vectors is the same after transformation, the direction
of angle may be inverted. All angle-preserving transformations are a ne
and invertible.
5.7.5 Orthogonal Transformations
Orthogonal is a term that is used to describe a matrix whose rows form
an orthonormal basis. Remember from Section 3.3.3 that the basic idea
is that the axes are perpendicular to each other and have unit length.
Orthogonal transformations are interesting because it is easy to compute
their inverse and they arise frequently in practice. We talk more about
orthogonal matrices in Section 6.3.
Translation, rotation, and reflection are the only orthogonal transfor-
mations. All orthogonal transformations are a ne and invertible. Lengths,
angles, areas, and volumes are all preserved; however in saying this, we
must be careful as to our precise definition of angle, area, and volume,
since reflection is an orthogonal transformation and we just got through
saying in the previous section that we didn't consider reflection to be an
angle-preserving transformation. Perhaps we should be more precise and
say that orthogonal matrices preserve the magnitudes of angles, areas, and
volumes, but possibly not the signs.
As Chapter 6 shows, the determinant of an orthogonal matrix is ±1.
5.7.6 Rigid Body Transformations
A rigid body transformation is one that changes the location and orienta-
tion of an object, but not its shape. All angles, lengths, areas, and volumes
are preserved. Translation and rotation are the only rigid body transfor-
mations. Reflection is not considered a rigid body transformation.
Rigid body transformations are also known as proper transformations.
All rigid body transformations are orthogonal, angle-preserving, invertible,
and a ne. Rigid body transforms are the most restrictive class of trans-
forms discussed in this section, but they are also extremely common in
practice.
The determinant of any rigid body transformation matrix is 1.
5.7.7 Summary of Types of Transformations
Table 5.1 summarizes the various classes of transformations. In this table,
a Y means that the transformation in that row always has the property
associated with that column. The absence of a Y does not mean “never”;
rather, it means “not always.”