Graphics Reference
In-Depth Information
where
is a unit vector that's the axis of rotation of the matrix and
θ
is the angle
v
of rotation. And we discussed how to recover the axis
from an
arbitrary rotation matrix (except that when the matrix was
I
, the axis could be any
unit vector and the angle was 0). The associated element
q
of
S
3
has
cos(
and the angle
θ
v
θ/
2
)
as its first coordinate and
sin(
θ/
2
)
v
as its last three coordinates. The case where
θ
2
)=
0, so the
last three entries are all zeroes. There
is
an ambiguity, however: When we found
the axis
=
0 and
is indeterminate presents no problem, because
sin(
θ/
v
and the angle
θ
we could instead have found
−
v
and
−θ
; those two
v
would have produced
q
instead of
q
. So our “inverse” to
K
really can produce
one of two opposite values, depending on choices made in the axis-and-angle
computation. To make all this concrete, we'll give pseudocode for a function
L
whose domain is
SO
(
3
)
and whose codomain is
pairs
of antipodal points in
S
3
;
L
will act as an inverse to
K
, in the sense that if
M
−
∈
SO
(
3
)
is a rotation matrix
are two elements of
S
3
, then
K
(
q
1
)=
K
(
and
L
(
M
)=
q
1
)=
M
.In
the pseudocode in Listing 11.3, neither
q1
nor
q2
is guaranteed to be a continuous
function of the entries of the matrix
m
.
{
q
1
,
−
q
1
}
−
Listing 11.3: Code to convert a rotation matrix to the two corresponding
quaternions.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
void RotationToQuaternion(Mat33 m, out Quaternion q1, out Quaternion q2)
{
// convert a 3x3 rotation matrix m to the two quaternions
// q1 and q2 that project to m under the map K.
if (
m
is the identity
)
{
q1 = Quaternion(1,0,0,0);
q2 = -q1;
return;
}
Vector3D omega;
double theta;
RotationToAxisAngle(m, omega, theta);
q1 = Quaternion(Math.cos(theta/2), Math.sin(theta/2)
*
omega);
q2 = -q1;
}
We now have a method for going from
S
3
to
SO
(
3
)
and for going from
SO
(
3
)
back to
pairs
of elements of
S
3
. To interpolate between rotations in
SO
(
3
)
,we'll
interpolate between points of
S
3
.
11.2.6.1 Spherical Linear Interpolation
Suppose we have two points
q
1
and
q
2
of the unit sphere, and that
q
1
q
2
, that
is, they're not antipodal. Then there's a unique shortest path between them, just
as on the Earth there's a unique shortest path from the North Pole to any point
except the South Pole. (There
is
a shortest path from the North to the South Pole;
the problem is that it's no longer unique—any line of longitude is a shortest path.)
We'll now construct a path
=
−
γ
that starts at
q
1
(i.e.,
γ
(
0
)=
q
1
), ends at
q
2
(i.e.,
(
1
)=
q
2
), and goes at constant speed along the shorter great arc between
them. This is called
spherical linear interpolation
and was first described for use
in computer graphics by Shoemake [Sho85], who called it
slerp.
There are three
steps.
γ
