Graphics Reference
In-Depth Information
-- -
+
ax by c
ab
t
=
.
2
2
We get our equation by substituting this t into
()
=+
S
PPPQP N
2
=+
2
t
.
2.2.3.5. Example.
We redo Example 2.2.3.3 using equation (2.14).
Solution.
In this case
-+-
2
xy
2
t
=
,
5
so that
-+-
2
xy
2
(
)
=
(
)
+
(
)
Sx y
,
x y
,
2
21
,
-
.
5
This equation simplifies to the same equation for S as before.
A final and more systematic way to compute reflections, one that is easier to
remember conceptually (given that one understands translations, rotations about the
origin, and the reflection about the x-axis), is based on the often useful general prin-
ciple that complicated problems should be solved by successively reducing them to
simpler ones until one arrives at a primitive problem whose solution is known.
Case 1
(The primitive problem). The equation for the reflection S
x
about the
x-axis.
This problem was solved in Example 2.2.3.2 above and we got equations (2.12).
Case 2.
The equation for a reflection about a line through the origin.
This case can be reduced to the Case 1 by first rotating the line to the x-axis, then
using the equation from Case 1, and finally rotating back.
Case 3
(The general case). The equation for a reflection about an arbitrary line.
By translating the line to a line through the origin we can reduce this case to Case
2, find the equation for that case, and then translate back.
The steps outlined in Cases 1-3 lead to the following characterization of a reflection:
2.2.3.6. Theorem.
Every reflection S in the plane can be expressed in the form
-11
STRSRT
=
,
x
where T is a translation, R is a rotation about the origin, and S
x
is the reflection about
the x-axis.
