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].
11.8 Exercises
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
 
 
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