Graphics Reference
In-Depth Information
2.2.7.5.
Show that any motion of the form
M
:
x
¢ =+
¢ =-
ax
by
2
2
y
bx
ay
,
where
a
+=
b
1
,
is a reflection about a line through the origin.
2.2.7.6.
Explain why
4
5
3
5
Mx
:
¢ =-
x
+
y
+
6
3
5
4
5
y
¢ =
x
+
y
+
1
is an orientation reversing motion that is
not
a reflection. On the other hand, M has
a fixed line. Find it.
2.2.7.7.
Show that every orientation reversing motion is a composite of a rigid motion and a
single reflection.
2.2.7.8.
Prove the following:
(a)
Every motion can be expressed as a composition of at most three reflections.
(b)
Every motion with one fixed point is the composite of at most two reflections.
Section 2.2.8
2.2.8.1.
Use frames to find a motion that sends the line
L
through
A
(2,1) and
B
(3,3) to the
x-axis and the point
A
to the origin.
2.2.8.2.
Use frames to find a motion which sends the line 2x + 3y = 5 to the line x -
2y = 3.
2.2.8.3.
Solve Exercise 2.2.7.3 using frames.
2.2.8.4.
Use frames to find the equations of the motion that sends the points
A
(-1,3),
B
(0,1),
and
C
(-2,1) to
A
¢(3,2),
B
¢(2,0), and
C
¢(1,2), respectively.
2.2.8.5.
Consider the lines
L
:
x
+=
39
y
and
L
:
3
x
-=
y
7
.
1
2
Assuming that the lines are oriented to the right, find the transformation that con-
verts from world coordinates to the coordinate system where
L
1
and
L
2
are the x- and
y-axis, respectively.
Section 2.3
2.3.1.
Find the equations of the similarity S that sends the points
A
(-1,3),
B
(0,1), and
C
(-2,1)
to
A
¢(0,6),
B
¢(2,2), and
C
¢(4,6), respectively.
