Graphics Reference
In-Depth Information
B.2
Geometric computations
A vector (a one-dimensional list of numbers) is often used to represent a point in space or a direction
and magnitude in space (e.g.,
Figure B.3
)
. A slight complication in terminology results because a direc-
tion and magnitude in space is also referred to as a
vector
. As a practical matter, this distinction is
usually not important. A vector in space has no position, only magnitude and direction. For geometric
computations, a matrix usually represents a transformation (e.g.,
Figure B.4
).
B.2.1
Components of a vector
A vector,
A
, with coordinates (
A
x
,
A
y
,
A
z
) can be written as a sum of vectors, as shown in
Equation B.17
,
in which
i
,
j
,
k
are unit vectors along the principal axes,
x
,
y
, and
z
, respectively.
A ¼ A
x
i þ A
y
j þ A
z
k
(B.17)
B.2.2
Length of a vector
1.0, and it
is said to be
normalized
. Dividing a vector by its length, forming a unit-length vector, is said to be
normalizing
the vector.
¼
q
A
jAj¼
2
x
þ A
2
y
þ A
2
z
(B.18)
y
(2,5)
(5,2 )
x
FIGURE B.3
A point and vector in two-space.
y
Q
cos
sin cos
sin
Q
p
P
x
FIGURE B.4
A matrix representing a rotation.
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