Transformation Operators - Essentials of Interactive Computer Graphics - page 186

Geometric properties. In general, a translation operator maintains the shape of
the input object. This includes the dimension (size) of the input edges, angles
between the edges, and direction of the edges. These geometric attributes are the
same before and after the operation.

( , )

Reversibility. For every translation operator T t x t y , there exists an inverse op-

erator T − 1

( , ) ( , )

t x t y where we can reverse the effect of T t x t y . The inverse trans-

form for the operator T ( t x , t y ) is simply

T − 1 ( t x , t y )= T ( − t x ,− t y ) .

The inverse operator is important when we want to undo the effect of a transform
operation.

Identity. When both t x and t y are equal to 0, the translation operator becomes a
no-operation operator, where it does not cause any changes to any position. The
knowledge of identity is important for initializing operators when programming
variables representing translation operators.

Extension to 3D. All of our discussions are valid in the third dimension, and
we can extend our results in a straightforward manner by simply including a third
parameter t z :

T ( t x , t y , t z ) ,

where the t z parameter describes the displacement in the z -axis direction.

8.2 The Scaling Operator

Implied parameters. As in
translation operators, and in
general for all transform oper-
ators for simplicity and read-
ability, the scaling factor pa-
rameters,

Scaling means change of size. The scaling operator

S ( s x , s y )

,areoftenleft
out and are implied in expres-
sions. For example,

( s x , s y )

scales a point V a from

V a = x a y a

V as = V a S ( s x , s y )
may be written as:

to a new position V as :

= x as y as

V as

= x a × s y ,

V as = V a S .

s x y a ×