Graphics Reference
In-Depth Information
Virtual
frame
buffer
y
y
Ray constructed
through pixel
center
x
x
z
Object
Space
World
Space
z
Screen Space
FIGURE 2.5
Transformation through spaces using ray casting.
Typically, when transforming a point in world space, the fourth component will be one. This means a
ðx; y; zÞ½x; y; z;
1
(2.4)
The basic transformations of rotate, translate, and scale can be kept in 4
4 transformation matrices.
The 4
4 matrix is the smallest matrix that can represent all of the basic transformations. Because it is a
square matrix, it has the potential for having a computable inverse, which is important for texture map-
ping and illumination calculations. In the case of rotation, translation, and nonzero scale transforma-
tions, the matrix always has a computable inverse. It can be multiplied with other transformation
matrices to produce compound transformations while still maintaining 4
4-ness. The 4
4 identity
2
3
2
3
2
3
x
y
1
1000
0100
0010
0001
x
y
1
4
5
¼
4
5
4
5
(2.5)
Typically in the literature, a point is represented as a 4
1 column matrix (also known as a
column
vector
) and is transformed by multiplying by a 4
4 matrix on the left (also known as
premultiplying
the column vector by the matrix), as shown in
Equation 2.5
in the case of the identity matrix. However,
some texts use a 1
4 matrix (also known as a
row vector
) to represent a point and transform it by
multiplying it by a matrix on its right (the matrix
postmultiplies
the row vector). For example,
postmultiplying a point by the identity transformation would appear as in
Equation 2.6
.
2
4
3
5
1000
0100
0010
0001
½xyz
1
¼½xyz
1
(2.6)
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