Graphics Reference
In-Depth Information
C
m
C
m
A
m
A
+
m
c
m
B
m
B
+
m
c
a
a
m
A
+
m
c
B
′
m
B
+
m
C
A
′
P
m
c
m
A
+
m
c
b
m
c
m
B
+
m
c
a
C
′
A
m
A
m
B
B
m
B
m
A
+
m
B
m
A
+
m
B
m
A
m
A
+
m
B
c
c
Fig. 11.10.
How the masses determine the positions of
A
,B
and
C
.
To summarize, given three masses
m
A
,m
B
and
m
C
located at
A
,
B
and
C
, the centroid
P
is given by
m
A
m
A
+
m
B
+
m
C
A
+
m
B
m
A
+
m
B
+
m
C
B
+
m
C
m
A
+
m
B
+
m
C
C
P
=
(11.9)
If we accept that
m
A
,m
B
and
m
C
can have any value, including zero, then
the barycentric coordinates of
P
will be affected by these values. For example,
if
m
B
=
m
C
= 0 and
m
A
=1,then
P
will be located at
A
with barycentric
coordinates (1, 0, 0). Similarly, if
m
A
=
m
C
=0and
m
B
=1,then
P
will be
located at
B
with barycentric coordinates (0, 1, 0). And if
m
A
=
m
B
=0and
m
C
=1,then
P
will be located at
C
with barycentric coordinates (0, 0, 1).
Now let's examine a 3D example as illustrated in Figure 11.11. The fig-
ure shows three masses 4, 8 and 12 and their equivalent mass 24 located at
(
x, y, z
).
The magnitude and coordinates of three masses are shown in the follow-
ing table, together with the barycentric coordinate
t
i
. The column headed
t
i
Y
4
12
24
8
y
z
x
X
Z
Fig. 11.11.
Three masses can be represented by a single mass located at the system's
centroid.