Transformation Operators - Essentials of Interactive Computer Graphics - page 192

Geometric properties. In general, a scale operator changes the shape of input
objects. In fact, changing the relative distances between input points is precisely
the main purpose of the scaling operator. When the scaling factors for all axes
are the same, the input and output shapes are the same; however, their sizes are
different. For example, the operator S doubles the size of any input while

maintaining the general shape (angles between edges and directions of edges).
The scaling operator with scaling factors of zero, S

( 2 , 2 )

( , )

0 0 , compresses all input

points to the origin. This is considered an undefined operator. In general, we

want to avoid zero scaling factors.

Reversibility. For every scaling operator with non-zero scaling factors, there
exists an inverse operator such that we can reverse the effect of the operator. The
inverse transform operator for S ( s x , s y ) is

S 1

1
s y

S − 1 =

s x ,

.

Identity. Unlike the translation operator where 0 displacement is the identity
operator, the scaling operator is a no-operation operator when both the s x and s y
scaling factors are 1 . 0.

Extension to 3D. For positive scaling factors, the extension to the third dimen-
sion is straightforward. In 3D, the scaling operator simply has the third parameter
S ,where s z is the scaling factor in the z -axis. Negative scaling factor in

3D space is an interesting way of creating mirrored objects.

( s x , s y , s z )

8.3 The Rotation Operator

Rotation means turning. The rotation operator

( θ )

R

rotates a point V a from

V a = x a y a

to a new position V ar :

= x ar y ar

V ar

= x a cos θ ,

θ − y a sin θ y a cos θ + x a sin