Graphics Reference
In-Depth Information
k
k
k
(
)
=
Â
Â
Â
¢
¢
=-=
aa
-
a
-
a
11 0.
i
i
i
i
i
=
0
i
=
0
i
=
0
But the vectors
v
1
-
v
0
,
v
2
-
v
0
,..., and
v
k
-
v
0
are linearly independent, so that
a
i
= a
i
¢ for i = 1,2, . . . ,k, which then also implies that a
0
= a
0
¢. This proves that the rep-
resentation for
w
is unique.
The rest of part (2) is left as an exercise.
Definition.
Using the notation in Theorem 1.7.4(1), the a
i
are called the
barycentric
coordinates
of
w
with respect to the points
v
i
. The point
1
(
)
vv
++ +
...
v
0
1
k
k
+
1
is called the
barycenter
of the simplex s.
1.7.5. Example.
Let
v
0
= (1,0),
v
1
= (4,0), and
v
2
= (3,5). We want to find the barycen-
tric coordinates (a
0
,a
1
,a
2
) of
w
= (3,1) with respect to these vertices.
Solution.
We must solve
(
+
(
)
+
()
=
()
a
10
,
a
40
,
a
35
,
31
,
0
1
2
for a
0
, a
1
, and a
2
. Since a
2
= 1 - a
0
- a
1
, we really have to solve only two equations in
two unknowns. The unique solutions are a
0
= 4/15, a
1
= 8/15, and a
2
= 1/5. The barycen-
ter of the simplex
v
0
v
1
v
2
is the point (8/3,5/3).
Theorem 1.7.4 shows that barycentric coordinates are another way to parame-
terize points, which is why that terminology is used. They are a kind of weighted sum
and are very useful in problems that deal with convex sets. In barycentric coordinates,
the point
w
in the definition would be represented by the tuple (a
0
,a
1
,...,a
k
). The
barycenter would have the representation
1
1
1
Ê
Ë
ˆ
¯
,
,...,
.
kk
++
1
1
k
+
1
Barycentric coordinates give information about ratios of volumes (or areas in
dimension 2). (For a general definition of volume in higher dimensions see Chapter
4.) Consider a simplex s =
v
0
v
1
···
v
k
and a point
w
in it. Let (a
0
,a
1
,...,a
k
) be the
barycentric coordinates of
w
. Let D be the volume of s and let D
i
be the volume of the
simplex with vertices
v
0
,
v
1
,...,
v
i-1
,
w
,
v
i+1
,...,
v
k
. See Figure 1.18.
D
D
i
1.7.6. Proposition.
a
i
=
.
Proof.
See [BoeP94].
Finally, barycentric coordinates are useful in describing linear maps between sim-
plices. Let f be a map from the set of vertices of a simplex s onto the set of vertices
