Graphics Reference
In-Depth Information
Mapping a vector is different from mapping a point. The justification of equation
(6.42) is that to map a vector one really should map its two “endpoints.” To put it
another way, if
p
and
q
are two points in
R
n
, then the direction vector
pq
transforms
to the direction vector
() ()
=
()
-
()
=
(
()
)
-
(
()
)
=
()
-
()
=
()
M
pq
M
M
q
M
p
TM
q
TM
p
M
q
M
p
M
q
.
0
0
0
0
0
It follows that translations leave vectors unchanged. For example, if M is defined by
equations
x x ym
y x yn
¢=
+
+
¢=
+
+ ,
then to transform a vector we can drop the translational part and it will transform
using the equations
x x y
y x y
¢=
+
¢=
+
.
Next, we look at how parameterizations and implicit representations of a space
X
in
R
n
change under a transformation
n
n
T
:
RR
Æ
.
Let
Y
= T(
X
). Assume that
X
is defined by a parameterization
n
p
:
AR
Æ
and is the set of zeros of
n
f
:
RR
Æ
.
Clearly,
Y
is parameterized by the composition T
°
p. If T has an inverse, then
Y
can
also be defined implicitly. In fact,
{
(
)
=
}
fT
1
-
Yq
=¢
()
q
0.
(6.43)
Figure 6.25 explains the validity of equation (6.43). The point
q
¢ belongs to
Y
if and
only if the point
q
= T
-1
(
q
¢) lies in
X
and satisfies the equation f(
q
) = 0.
Figure 6.25.
Transforming an implicit
representation.
