Graphics Reference
In-Depth Information
we continue, here is a summary of the properties of the transformation operators
we have studied:
Operator
Inverse
Identity
Maintains Shape
T
(
t
x
,
t
y
)
T
(
−
t
x
,−
t
y
)
T
(
,
)
0
0
Ye s
1
s
x
,
1
s
y
)
S
(
s
x
,
s
y
)
S
(
S
(
1
,
1
)
No
R
(
θ
)
R
(
−
θ
)
R
(
0
)
Ye s
8.5
Some Mathematics of the Transform Operators
This section is not meant to present a complete mathematical background for the
transformation operators. Instead, we list some relevant facts and properties for
these operators. Our discussion is limited to vectors in two or three dimensions
and 4
4 matrices representing the well-defined affine transformations. For a
discussion of more involved operations with vectors, see Appendix B. To learn
more about the mathematics of vectors and matrices, see a classic textbook on
linear algebra.
×
Triangles.
A triangle is the
simplest geometric shape with
area that can be defined by
vertices, and thus it is the
most straightforward to pro-
cess. Triangles have many el-
egant properties; for example,
the three vertices of a triangle
are guaranteed to form a flat
plane.
We have seen that it is straightforward to represent vertex positions as vectors.
We have also seen that the transformation operators are well-defined mathematical
operations that manipulate vectors. The mathematical foundation allows custom-
designed hardware processors to efficiently implement these operations. Today,
virtually all commercial graphics hardware is based on transforming and process-
ing vertices. As a direct result, all popular graphics APIs are also defined based
on processing vertices. To simplify hardware implementation, only triangles are
explicitly processed by the hardware. In all cases, the vast majority of graphi-
cal objects we encountered are defined based on collections of triangles. For this
reason, we will concentrate on working with a collection of vertices (or vectors).
Typically, the vertices represent a collection of triangles, which in turn represent
some meaningful geometrical shape (e.g., a sphere, a car).
8.5.1
A Word about Vectors
In the beginning of this chapter, when we first introduced transformation opera-
tors, we said that we can represent a vertex position
V
a
:
V
a
=(
x
a
,
y
a
,
z
a
)
with the vector representation
V
a
:
V
a
=
x
a
y
a
z
a
.
Search WWH ::
Custom Search