Game Development Reference
In-Depth Information
third basis vector was chosen to be any other vector that didn't lie in the
plane spanned by
p
and
q
(for example, the vector
c
), then the basis would
be linearly independent and have full rank. If a set of basis vectors are
linearly independent, then it is not possible to express any one basis vector
as a linear combination of the others.
So a set of linearly dependent vectors is certainly a poor choice for
a basis. But there are other more stringent properties we might desire
of a basis. To see this, let's return to coordinate space transformations.
Assume, as before. that we have an object whose basis vectors are
p
,
q
,
and
r
, and we know the coordinates of these vectors in world space. Let
b
= [b
x
,b
y
,b
z
] be the coordinates of some arbitrary vector in body space,
and
u
= [u
x
,u
y
,u
z
] be the coordinates of that same vector, in upright
space. From our robot example, we already know the relationship between
u
and
b
:
u
x
= b
x
p
x
+ b
y
q
x
+ b
z
r
x
,
u
= b
x
p
+ b
y
q
+ b
z
r
, or equivalently, u
y
= b
x
p
y
+ b
y
q
y
+ b
z
r
y
,
u
z
= b
x
p
z
+ b
y
q
z
+ b
z
r
z
.
Make sure you understand the relationship between these equations and
Equation (3.1) before moving on.
Now here's the key problem: what if
u
is known and
b
is the vector
we're trying to determine? To illustrate the profound difference between
these two questions, let's write the two systems side-by-side, replacing the
unknown vector with “?”:
?
x
= b
x
p
x
+ b
y
q
x
+ b
z
r
x
,
u
x
=?
x
p
x
+?
y
q
x
+?
z
r
x
,
?
y
= b
x
p
y
+ b
y
q
y
+ b
z
r
y
,
u
y
=?
x
p
y
+?
y
q
y
+?
z
r
y
,
?
z
= b
x
p
z
+ b
y
q
z
+ b
z
r
z
,
u
z
=?
x
p
z
+?
y
q
z
+?
z
r
z
.
The system of equations on the left is not really much of a “system” at
all, it's just a list; each equation is independent, and each unknown quan-
tity can be immediately computed from a single equation. On the right,
however, we have three interrelated equations, and none of the unknown
quantities can be determined without all three equations. In fact, if the ba-
sis vectors are linearly dependent, then the system on the right may have
zero solutions (
u
is not in the span), or it might have an infinite number
of solutions (
u
is in the span and the coordinates are not uniquely de-
termined). We hasten to add that the critical distinction is not between
upright or body space; we are just using those spaces to have a specific
example. The important fact is whether the known coordinates of the
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