Graphics Reference
In-Depth Information
Inline Exercise 11.4:
(a) What's
?
v
×
v
(b) Show that
J
=
0
as expected.
(c) Suppose that
v
is a unit vector perpendicular to
v
v
. Explain why
v
is
v
v
×
perpendicular to both, and why
(
v
)=
−
v
.
v
×
v
×
The rotation matrix we seek is then
)
J
2
v
M
=
I
+sin(
θ
)
J
+(
1
−
cos
θ
.
(11.17)
v
From Inline Exercise 11.4, it's clear that
M
=
.Andif
v
is perpendicular
v
v
to
v
, then
)
J
2
v
Mv
=
Iv
+sin(
θ
)
J
v
+(
1
−
cos
θ
v
(11.18)
v
=
v
+sin(
θ
)
v
+(
1
−
cos
θ
)(
(
v
))
(11.19)
v
×
v
×
v
×
=
v
+sin(
θ
)
v
+(
1
−
cos
θ
)(
−
v
)
(11.20)
v
×
=sin(
θ
)
v
+cos(
θ
)(
v
)
,
(11.21)
v
×
cos(
u
)
v
1
sin(
u
)(
v
3
v
)
which is just the rotation of
v
in the plane perpendicular to
by an angle
θ
. Since
v
M
does the right thing to
, it must be the right
matrix, as per the Transformation Uniqueness principle (see Figure 11.2).
and to vectors perpendicular to
v
v
v
3
v
v
v
In coordinate form, it's
u
⎡
⎤
0
−
v
z
v
y
⎣
⎦
M
=sin
θ
0
−
v
x
(11.22)
v
z
−
v
y
0
v
x
Figure 11.2: When
v
is orthogo-
nal to
⎡
⎤
y
z
−
v
−
v
v
x
v
y
v
z
v
x
v
×
v
form a basis
for the plane perpendicular to
v
,
v
and
⎣
⎦
+
I
,
z
x
+(
1
−
cos
θ
)
v
x
v
y
−
v
−
v
(11.23)
v
.
v
y
v
z
x
y
v
y
v
z
−
v
−
v
v
z
v
x
where we've used the fact that
is a unit vector to simplify things a little. But the
earlier form is far easier to program correctly.
v
As we said earlier, it's a theorem that every rotation of 3-space has an axis (i.e.,
a vector that it leaves untouched). We can use Rodrigues' formula to recover the
axis from the matrix. We'll follow the approach of Palais and Palais [PP07].
We know every rotation matrix has an axis
and an amount of rotation,
θ
,
v
about
; Rodrigues' formula tells us the matrix must be
v
)
J
2
v
M
=
I
+sin(
θ
)
J
v
+(
1
−
cos
θ
(11.24)
for some unit vector
.
The trace of this matrix (the sum of the diagonal entries) is
and some angle
θ
v
)
J
2
v
tr
(
M
)=
tr
(
I
+sin(
θ
)
J
+(
1
−
cos
θ
)
v
)
tr
(
J
2
v
=
tr
(
I
)+sin(
θ
)
tr
(
J
v
)+(
1
−
cos
θ
)
2
2
2
=
3
+(
1
−
cos
θ
)(
−
2
(
x
+
y
+
z
))
v
v
v
=
3
+(
1
−
cos
θ
)(
−
2
)
=
1
+
2
cos
θ
,
