Graphics Reference
In-Depth Information
v
3
v
1
v
2
v
2
v
1
v
0
v
1
v
0
v
0
v
0
s
1
= v
0
v
1
is the
line segment
from v
0
to v
1
s
2
= v
0
v
1
v
2
is the
solid triangle
s
3
= v
0
v
1
v
2
v
3
is
the solid tetrahedron
s = v
0
(d)
(a)
(b)
(c)
Figure 1.17.
Some simplices.
simplex. In general,
R
n
contains at most n-dimensional simplices because it is not
possible to find j linearly independent points in
R
n
for j > n + 1. Also, a simplex depends
only on the set of vertices and not on their ordering. For example,
v
0
v
1
=
v
1
v
0
. K-
simplices are the simplest kind of building blocks for linear spaces called simplicial
complexes, which are defined in Chapter 6, and they play an important role in alge-
braic topology. They have technical advantages over other regularly shaped regions
such as cubes. In particular, their points have a nice representation as we shall show
shortly in Theorem 1.7.4.
1.7.3. Lemma.
(1) The set aff({
v
0
,
v
1
,...,
v
k
}) consists of the points
w
that can be written in the
form
k
k
Â
a
Â
wv
=
,
where
a
=
1
(1.25)
ii
i
i
=
0
i
=
0
(2) The set conv({
v
0
,
v
1
,...,
v
k
}) consists of the points
w
that can be written in the
form
k
k
Â
a
Â
Œ
[]
wv
=
,
where
a
01
,
and
a
=
1
.
ii
i
i
i
=
0
i
=
0
Proof.
To prove (1), let
k
k
Ï
Ó
¸
˛
Â
a
Â
S
=
v
a
=
1
.
ii
i
i
=
0
i
=
0
