Graphics Reference
In-Depth Information
which are used as scalars to multiply sequences of control points to create
a parametric curve. Furthermore, these terms sum to unity, therefore they
are also another form of barycentric coordinates. The only difference between
these terms and the others described above is that they are controlled by a
common parameter
t
. Another property of Bezier curves and patches is that
they are constrained within the convex hull formed by the control points,
which is also a property of barycentric coordinates.
11.9 Summary
To summarize, barycentric coordinates are regularly used to determine:
1. How a value is divided into various ratios. For example, a scalar
A
is divided
into the ratios
r
:
s
:
t
using
r
r
+
s
+
t
A,
s
r
+
s
+
t
Aand
t
r
+
s
+
t
A
2. The mid-point between two points
A
and
B
:
P
=
1
2
A
+
1
2
B
3. The centroid of triangle ∆
ABC
:
P
=
1
3
A
+
1
3
B
+
1
3
C
4. A point on a line through two points
A
and
B
:
P
=(1
−
t
)
A
+
t
B
5. Whether a point is inside or outside triangle ∆
ABC
:
P
=
r
A
+
s
B
+
t
C
P
is inside or on the boundary of triangle ∆
ABC
when 0
≤
r, s, t
≤
1,
otherwise it is outside.
6. Whether a point is inside a tetrahedron
P
1
,P
2
,P
3
,P
4
:
P
=
r
P
1
+
s
P
2
+
t
P
3
+
u
P
4
P
is inside tetrahedron
P
1
,P
2
,P
3
,P
4
when 0
≤ r, s, t, u ≤
1, otherwise it is
outside.
7. Centres of gravity:
i
=1
m
i
x
i
n
i
=1
m
i
y
i
n
i
=1
m
i
z
i
n
n
n
n
x
=
y
=
z
=
i
=1
m
i
i
=1
m
i
i
=1
m
i
where
m
i
is a mass located at
x
i
.
