Graphics Reference
In-Depth Information
3
2
1
2
(
)
++
(
)
-
y
¢=
x
+
3
y
11.
The form of the solution to Example 2.2.2.5 generalizes to
2.2.2.6. Theorem.
The equations for a rotation R about a point
p
= (a,b) through
an angle q are
(
)
(
)
xxa
¢=
-
cos
q
--
yb
sin
q
+
a
(
)
(
)
yxain yb sb
¢=
-
s
q
+-
c
q
+
.
Proof.
Exercise.
Three interesting properties of rotations are
2.2.2.7. Proposition.
(1) The only fixed point of a rotation that is not the identity map is its
center.
(2) All rotations change the slope of a line unless the rotation is through an angle
of 0 or p.
(3) Only the rotations through an angle of 0 or p have a fixed line.
Proof.
We shall only give a proof of (2). The proof of (1) is left as an exercise and
(3) is an immediate consequence of (2).
We already know from Proposition 2.2.1.2 that translations do not change
slopes. Therefore it suffices to prove (2) for rotations R about the origin. Let
L
be a line defined by the equation ax + by = c. If R is a rotation through an angle
q and
L
¢
= R(
L
), then substituting for x and y using the equations for R
-1
we get
that
(
)
+-
(
)
=
ax
cos
q
+
y
sin
q
b x
sin
q
+ y cos
q
c
is an equation for
L
¢. The proof of (2) in the special case where either
L
or
L
¢ is ver-
tical is easy and is left as an exercise. In the rest of the discussion we assume that
slopes are defined. It follows that the slope for
L
¢
is
b
sin
q
-
+
a c
os
q
.
a
sin
q
b c
os
q
But this quotient can never equal the slope of
L
which is -a/b unless sin q=0, that is,
q=0 or p. (Simply set the two expressions equal and simplify the resulting equation
to get b
2
sin q=-a
2
sin q.) This proves the result.
