Graphics Reference
In-Depth Information
1.6.5.
Show that the angle between oriented hyperplanes (
X
,s) and (
Y
,t) is well defined.
Specifically, show that it does not depend on the choice of the normal vectors
v
n
and
w
n
for
X
and
Y
, in the definition.
1.6.6.
Let
L
be an oriented line and let
p
and
q
be two points on
L
. Prove that
pq
=
0,
if
if the vector
p
=
q
,
=
=-
pq
,
pq
induces the same orientation on
L
,
and
pq
,
if
pq
induces the opposite orientation on L
.
1.6.7.
Let (
V
,s) and (
W
,t) be two oriented n-dimensional vector spaces and let T :
V
Æ
W
be
a nonsingular linear transformation. Show that T is orientation preserving if
[
()( )
( )
]
t=
TT
v
,
v
,...,
T
n
v
1
2
for any
one
ordered bases (
v
1
,
v
2
,...,
v
n
) of
V
with the property that s=[
v
1
,
v
2
,...,
v
n
].
Section 1.7
Show that each halfplane in
R
n
is convex.
1.7.1.
1.7.2.
Show that if
X
1
,
X
2
,...,
X
k
are convex sets, then their intersection is convex.
1.7.3.
If
X
is convex, show that conv(
X
) =
X
.
1.7.4.
Show that conv({
p
0
,
p
1
}) = [
p
0
,
p
1
].
1.7.5.
Let s be the two-dimensional simplex defined by the vertices
v
0
= (-2,-1),
v
1
= (3,0), and
v
2
= (0,2). The points of s can be described either with Cartesian or barycentric
coordinates (with respect to the vertices listed in the order given above).
(a)
Find the Cartesian coordinates of the point
p
whose barycentric coordinates are
1
4
5
12
1
3
Ê
Ë
ˆ
¯
,
,
.
(b)
Find the barycentric coordinates of the point
q
whose Cartesian coordinates are
(0,0).
1.7.6.
Show that the simplicial map from the 1-simplex [2,5] to the 1-simplex [3,7] that sends
2 to 3 and 5 to 7 agrees with the “standard” linear map between the intervals, namely,
4
3
1
3
.
()
=+
gx
x
1.7.7.
Generalize Exercise 1.7.6 and show that the simplicial map from [a,b] to [c,d] agrees
with the standard linear map.
Section 1.8
1.8.1.
Let
12
32
.
A =
Ê
ˆ
˜
Á
Find a matrix P so that P
-1
AP is a diagonal matrix.
