Transformation Operators - Essentials of Interactive Computer Graphics - page 182

One simple way to describe an interactive computer graphics application is that
it is a program that allows the user to interactively manipulate geometric objects.
Indeed, as we have experience from the ball-shooting program, where we create
balls of different sizes (scales) and watch them fall (move) under the influence
of gravity. Here we see that by “manipulate,” we mean move, scale, rotate, and
so on. In computer graphics, we refer to these manipulation operations as the
transformation operations. It is interesting that in this entire topic, we only work
with three different transformation operators: translate , scale ,and rotate .These
three basic operators are sufficient for most real-world complex graphical envi-
ronments. Even more remarkable is the fact that these three operators are very
simple and that they operate on points. In this chapter, we introduce the mathe-
matical tools borrowed from linear algebra to help us describe these operators.
In this topic, we are not interested in the general theories behind the trans-
formation operators. We will not present the general theories nor any proof as-
sociated with the tools we learn. We are only interested in learning the facts and
experiencing how to apply these facts in computer graphics applications. Read-
ers interested in the theories behind the materials presented in this chapter should
consult an introductory linear algebra textbook.

Vector representation.
V = xy

In our discussion, we express all coordinate values in two dimensions. For
example, a vertex position

represents the vertex position
V =( x , y ) by the vector V .
As will be discussed in Sec-
tion 8.5, this representation
supports convenient mathe-
matics operations for transfor-
mation of vertex positions.

V =( x , y ) ,

is expressed as a vector:

= xy .
For simplicity, the third dimension of z

V

0 is left out. Although all our examples
are given in two dimensions, the results we derive are generally applicable in three
dimensions. At the end of the discussion for each transformation operator, we
present a brief discussion on how to generalize our results to the third dimension.

=

8.1 The Translation Operator

Translation means moving. The translation operator:

T ( t x , t y )

moves a point V a from

V a = x a y a

to a new position V at :

= x at y at

V at

= x a + t y ,

t x y a +