Graphics Reference
In-Depth Information
Chapter 10
Two Dimensions
As you saw in Chapters 2 and 6, when we think about taking an object for which
we have a geometric model and putting it in a scene, we typically need to do three
things:
Move
the object to some location,
scale
it up or down so that it fits well
with the other objects in the scene, and
rotate
it until it has the right orientation.
These operations—translation, scaling, and rotation—are part of every graphics
system. Both scaling and rotation are
linear transformations
on the coordinates
of the object's points. Recall that a linear transformation,
T
:
R
2
R
2
,
→
(10.1)
T
(
w
)
for any two vectors
v
and
w
in
R
2
,
is one for which
T
(
v
+
α
w
)=
T
(
v
)+
α
and any real number
. Intuitively, it's a transformation that preserves lines and
leaves the origin unmoved.
α
Inline Exercise 10.1:
Suppose
T
is linear. Insert
=
1 in the definition of
linearity. What does it say? Insert
v
=
0
in the definition. What does it say?
α
Inline Exercise 10.2:
When we say that a linear transformation “preserves
lines,” we mean that if
is a line, then the set of points
T
(
)
must also
lie in
some line. You might expect that we'd require that
T
(
)
actually
be
a line, but
that would mean that transformations like “project everything perpendicularly
onto the
x
-axis” would not be counted as “linear.” For this particular projection
transformation, describe a line
such that
T
(
)
is contained in a line, but is not
itself a line.
221
