Linear Algebra

Topic: Orthonormal Matrices 289
� � −b
a
� � a
b
� �
x t
↦−→
y
� �
x cos θ − y sin θ
x sin θ + y cos θ
The second matrix above represents a reflection of the plane through the line
bisecting the angle between �e1 and t(�e1). (This picture shows �e1 reflected up
into the first quadrant and �e2 reflected down into the fourth quadrant.)
� � a
b
� � b
−a
� �
x t
↦−→
y
� �
x cos θ + y sin θ
x sin θ − y cos θ
Note that in this second case, the right angle from �e1 to �e2 has a counterclockwise
sense but the right angle between the images of these two has a clockwise sense,
so the sense gets reversed. Geometers speak of a distance-preserving map as
direct if it preserves sense and as opposite if it reverses sense.
So, we have characterized the Euclidean study of congruence into the consideration
of the properties that are invariant under combinations of (i) a rotation
followed by a translation (possibly the trivial translation), or (ii) a reflection
followed by a translation (a reflection followed by a non-trivial translation is a
glide reflection).
Another idea, besides congruence of figures, encountered in elementary geometry
is that figures are similar if they are congruent after a change of scale.
These two triangles are similar since the second is the same shape as the first,
but 3/2-ths the size.
P1
P2
P3
From the above work, we have that figures are similar if there is an orthonormal
matrix T such that the points �q on one are derived from the points �p by �q =
(kT)�v + �p0 for some nonzero real number k and constant vector �p0.
Although many of these ideas were first explored by Euclid, mathematics is
timeless and they are very much in use today. One application of rigid motions
is in computer graphics. We can, for example, take this top view of a cube
and animate it by putting together film frames of it rotating.
Q1
Q2
Q3