Graphics Reference
In-Depth Information
(1) T has no fixed points.
(2) T takes lines to lines with the same direction vector (or slope, in the case of
the plane).
(3) The only lines fixed by T are those with direction vector
v
. In the case of the
plane, the only lines fixed by T are those whose slope is the same as the slope
of one of their direction vectors.
Proof.
(1) and (2) are left as exercises for the reader. To prove (3), consider a line
L
through a point
p
0
with direction vector
w
. If T fixes
L
, then T maps a point
p
0
+ t
w
on
L
to another point on
L
that will have the form
p
0
+ s
w
. Therefore,
+=
( )
=++
pw pw
pwv
s
T t
t
0
0
,
0
and so
w
is a multiple of
v
. The converse is just as easy.
In case of the plane, assume that the line
L
fixed by T is defined by the equation
ax
+=.
by
c
(2.4)
The line L has slope -a/b (the case of a vertical line where b is zero is left as an exer-
cise for the reader). If
v
= (h,k), then the slope of
v
is k/h. Choose a point (x,y) on
L
.
Since T(x,y) = (x + h,y + k) is assumed to lie on
L
, that point must also satisfy equa-
tion (2.4), that is,
(
)
++
(
)
=
ax h
+
by k
c
.
Using the identity (2.4) in this last equation implies that ah + bk = 0. This shows that
k/h =-a/b and we are done.
2.2.2
Rotations in the Plane
Another intuitively simple motion is a rotation of the plane.
Definition.
Let qŒ
R
. A map R :
R
2
Æ
R
2
of the form R(r,a) = (r,a+q), where points
have been expressed in polar coordinates, is called a
rotation about the origin through
an angle
q.
See Figure 2.3. Using polar coordinates was an easy way to define rotations about
the origin, but is not convenient from a computational point of view. To derive the
equations for a rotation R in Cartesian coordinates, we use the basic correspondence
between the polar coordinates (r,a) and Cartesian coordinates (x,y) for a point
p
:
xr
yr
=
=
cos
sin
a
a
(2.5)
Let R(x,y) = (x¢,y¢). Since R(r,a) = (r,a+q),
