Game Development Reference
In-Depth Information
The numeric coordinates of a vector with respect to a given basis are the
coe
cients in the expansion of that vector as a linear combination of the
basis vectors. For example,
a
= a
x
p
+ a
y
q
.
The reason the dot product doesn't “sift out” the coordinates in
Fig-
placement parallel to
p
. To visualize this, imagine that we increased a
x
while holding a
y
constant. As
a
moves to the right and slightly upwards,
its projection onto
q
, which is measured by the dot product, increases.
The problem is that the basis vectors are not perpendicular. A set of
basis vectors that are mutually perpendicular is called an orthogonal basis.
When the basis vectors are orthogonal, the coordinates are uncoupled. Any
given coordinate of a vector
v
can be determined solely from
v
and the
corresponding basis vector. For example, we can compute v
x
knowing only
p
, provided that the other basis vectors are perpendicular to
p
.
Although we won't investigate it further in this topic, the idea of or-
thonormal basis is a broadly powerful one with applications outside our
immediate concerns. For example, it is the idea behind Fourier analysis.
If it's good when basis vectors are orthogonal, then it's best when they
all have unit length. Such a set of vectors are known as an orthonormal
basis. Why is the unit length helpful? Remember the geometric definition
of the dot product:
a
p
is equal to the signed length of
a
projected onto
p
, times the length of
p
. If the basis vector doesn't have unit length, but it
is perpendicular to all the others, we can still determine the corresponding
coordinate with the dot product; we just need to divide by the square of
the length of the basis vector.
In an orthonormal basis, each coordinate of a vector
v
is the signed dis-
placement
v
measured in the direction of the corresponding basis vector.
This can be computed directly by taking the dot product of
v
with that
basis vector.
Search WWH ::
Custom Search