Homework 5
1. If
‘
and
m
are lines which are not parallel, prove that there exists a point
P
and an angle
θ
so that
ρ
P,θ
(
‘
) =
m
.
Solution:
Let
P
be the unique point of intersection of
‘
and
m
and let
θ
be the directed
angle from
‘
to
m
. Then
ρ
P,θ
maps
‘
to
m
. To see this, note that
ρ
P,θ
sends
P
to
P
and
for any other point
Q
on
‘
, it is mapped to a point
Q
0
on
m
. Since isometries send lines to
lines, it then follows that
ρ
P,θ
(
‘
) =
m
as desired.
2. For any two lines
‘,m
show that there is a line
n
so that
σ
n
(
‘
) =
m
.
Solution:
If
‘,m
are parallel, choose a line
n
parallel to
‘,m
so that the directed distance
from
‘
to
n
is exactly
1
2
the directed distance from
‘
to
m
. Now
σ
n
(
‘
) =
m
(to check this,
just note that any two points on
‘
map to
m
). If
‘,m
are not parallel and intersect ath the
point
P
with directed angle from
‘
to
m
of
θ
, then let
n
be the line passing through
P
so
that the directed angle from
‘
to
n
is
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 RWK

Click to edit the document details