Graphics Reference
In-Depth Information
interest in them in physics (and some related interest in graphics), with recent
developments being given the name
geometric algebra
[HS84, DFM07].
Exercise 11.1:
We computed the matrix for reflection through the line determined
by the unit vector
u
by reasoning about dot products. But this reflection is also
identical to rotation about
u
by 180
◦
. Use the axis-angle formula for rotation to
derive the reflection matrix directly.
Exercise 11.2:
In
R
n
, what is the matrix for reflection through the subspace
spanned by
e
1
,
,
e
k
, the first
k
standard basis vectors? In terms of
k
, tell whether
this reflection is orientation-
preserving
or
reversing
.
Exercise 11.3:
Write down the matrices for rotation by 90
◦
in the
xy
-plane and
rotation by 90
◦
in the
yz
-plane. Calling these
M
and
K
, verify that
MK
...
=
KM
,
thus establishing that in general, if
R
1
and
R
2
are elements of
SO
(
3
)
, it's not true
that
R
1
R
2
=
R
2
R
1
; this is in sharp contrast to the set
SO
(
2
)
of 2
×
2 rotation
matrices, in which any two rotations commute.
Exercise 11.4:
In Listing 11.2, we have a condition “if
” which
handles the special case of very-large-angle rotations by picking a nonzero column
v
of
M
+
I
as the axis. If
θ
is near
π
, then
v
will not quite be parallel to an
axis. Explain why
v
+
Mv
will be much more nearly parallel to the axis. Adjust
the code in Listing 11.2 to apply this idea repeatedly to produce a very good
approximation of the axis.
Exercise 11.5:
Consider the parameterization of rotations by Euler angles,
with
θ
is not exactly
π
θ
=
π/
2. Show that simultaneously increasing
φ
and decreasing
ψ
by the
same amount results in no change in the rotation matrix at all.
Exercise 11.6:
The second displayed matrix in Equation 11.23 is the square of
the first
(
J
v
)
; it's also symmetric. This is not a coincidence. Show that the square
of a skew-symmetric matrix is always symmetric.
Exercise 11.7:
Find the eigenvalues and all real eigenvectors for
J
v
.Dothe
same for
J
2
v
.
Exercise 11.8:
Suppose that
A
isarotationmatrixin
R
3
.
(a) How many eigenvalues does a 3
3matrixhave?
(b) Show that the only real eigenvalue that a rotation matrix can have is
×
±
1. Hint:
A rotation preserves length.
(c) Recall that for a real matrix, nonreal eigenvalues come in pairs: If
z
is an
eigenvalue, so is
z
. Use this to conclude that
A
must have either one or three real
eigenvalues.
(d) Use the fact that if
z
is a nonzero complex number, then
zz
0, and the fact that
the determinant is the product of the eigenvalues to show that if
A
has a nonreal
eigenvalue, it also has a real eigenvalue which much be 1, and that if
A
has only
real eigenvalues, at least one of them must be 1.
(e) Conclude that since 1 is always an eigenvalue of
A
, there's always a nonzero
vector
v
with
Av
=
v
, that is, the rotation
A
has an axis.
Exercise 11.9:
The skew-symmetric matrix
J
>
associated to a vector
is the
v
v
matrix for the linear transformation
v
→
v
×
v
.
(a) Show that every 3
×
3 skew-symmetric matrix
S
represents the cross product
with some vector
, that is, describe an inverse to the mapping
v
→
J
.
v
v