Graphics Reference
In-Depth Information
Figure 8.9 shows an example of what we may expect, intuitively, from a scal-
ing operation with scaling factors of
Original
Rectangle
(
3
,
0
.
5
)
. In this case, the 2
×
3 rectangle is
scaled with respect to a reference position at the lower-left corner of the rectangle
at
R o
, resulting in the rectangle ( R o ). We will describe how to accomplish this
kind of transformation in the next chapter after we have a good understanding of
the basic transformation operators.
(
1
,
5
)
Reference=(1,5)
Figure 8.9. Example of
an “intuitive” scaling oper-
ation.
Negative Scaling Factors: Reflections
Applying a scaling operator with negative scaling factor has the effect of reflecting
the input points across axes. For example, if we apply a scale operator with s x =
1and s y =
1(or S
(
1
,
1
)
) on the point V a =(
3
,
4
)
,weget
V as = x as y as
=
V a S
(
s x ,
s y )
= x a ×
s y
s x y a ×
= 3
1
×−
14
×
=
34 .
Y-axis
S(-1,1)
This result is illustrated in Figure 8.10. We see that with an s x of
1, the input
point V a is reflected across the y -axis and not the x -axis. Reflection is a special
case of the scaling operator, and we have learned that the scaling operator can be
applied to multiple points. Figure 8.11 shows the results of reflecting four vertices
of a rectangle:
V as (-3,4 )
V a (3,4)
X-ax is
Origin
Figure 8.10. Effect of
negative scaling factor.
=(
,
) ,
V a
3
4
V b =(
1
,
4
) ,
Input points:
=(
,
) ,
V c
1
1
V d =(
3
,
1
) .
We have already seen that scaling by S
(
1
,
1
)
simply flipped the x -coordinate
value of V a to V as =(
. In this example, we see that the flipping applies to
the rest of the vertex positions:
3
,
4
)
V as =
3
,
4
) ,
V bs =
1
,
4
) ,
Output points:
V cs =
1
,
1
) ,
V ds =
,
) .
3
1
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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