Graphics Reference
In-Depth Information
11.2.6 Rotations and the 3-Sphere
The set SO ( 3 ) of all 3
×
3 rotation matrices can be difficult to understand. It is,
in some sense, a subset of R 9 : Just read the nine entries of the matrix M in order
to get the point in R 9 that corresponds to M . In a web extra, we give consider-
able detail on this set and its properties. Here, we'll give just the essentials. The
main tool used to understand SO ( 3 ) , to make computations involving SO ( 3 ) more
numerically robust, and to let us reason about operations in SO ( 3 ) (like interpo-
lation) by means of a familiar space, is S 3 , the 3-sphere, or the set of all points
wxyz T in 4-space whose distance from the origin is 1. We've shuffled the
coordinates on purpose to make some of what we say below involve less shuffling.
We'll also, in this section, talk about points of S 3 , but we'll always write them as
vectors so that we can form linear combinations of them.
Figure
11.3:
Wrapping
a
line
onto a circle.
Just as you can wrap a line segment into a circle (joining the two bound-
ary points into one point of the circle, as in Figure 11.3), or wrap a disk into a
sphere (with the whole boundary circle becoming one point of the sphere, as in
Figure 11.4), you can wrap a solid ball in 3-space into a 3-sphere, by collaps-
ing the whole boundary sphere to a point. To do so, you must work in the fourth
dimension, but the idea is to reason about the 3-sphere by analogy.
For instance, if we take two perpendicular unit vectors u and v in the unit cir-
cle and construct all points of the form cos(
) v , these points cover the
whole circle (see Figure 11.5). Similarly, in the 2-sphere, if we have two perpen-
dicular unit vectors, their cosine-sine combinations form a great circle, that is,
the intersection of the sphere with a plane through its center (see Figure 11.6). In
fact, the same thing is true for the 3-sphere as well: Cosine-sine combinations of
perpendicular vectors traverse a great circle on the 3-sphere. And the arc of this
circle from u to v (i.e, from
θ
) u +sin(
θ
θ
= 0to
π/
2) is the shortest path between them, just
as in the lower dimensions.
There's a mapping from S 3 to SO ( 3 ) , given by
K : S 3
SO ( 3 ):
(11.32)
Figure 11.4: Wrapping a disk
onto a sphere; all points of the
circular edge of the disk are sent
to the North Pole.
a
b
c
d
a 2 + b 2
c 2
d 2
2 ( bc
ad )
2 ( ac + bd )
.
2 ( ad + bc ) a 2
b 2 + c 2
d 2
2 ( cd
ab )
2 ( bd
ac )
2 ( ab + cd ) a 2
b 2
c 2 + d 2
(11.33)
The map K has several nice properties.
cos(
u
) v
1
sin(
u
) w
S 3 ,
•s almost one-to-one. In fact, it's a two-to-one map; for any q
w
q ) , as you can see by looking at the formula.
• Great circles on S 3 are sent, by K , to geodesics (paths of shortest length) in
SO ( 3 ) .
K ( q )= K (
v
u
K ( 1000 T )= I .
This mapping arises from a definition of a kind of “multiplication” on R 4 ,
closely analogous to the way we can treat points of R 2 as complex numbers
and multiply them together. The multiplication on R 4 is not commutative, which
causes some difficulties, but otherwise it's closely analogous to multiplication of
complex numbers. The set R 4 , together with this multiplication operation, is called
the quaternions (which is why we use a boldface q for a typical element of S 3 ).
Figure 11.5: The set of all cosine-
sine combinations of v and w
wraps around the whole circle.
 
 
 
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