
534 Chapter 10 Standard Objects
Any point on the triangle may be represented in plane coordinates using Equation
(10.1). More common, though, is to write the points using barycentric coordinates:
X = b
0
P
0
+ b
1
P
1
+ b
2
P
2
where b
i
∈ [0, 1] for all i and b
0
+ b
1
+ b
2
= 1. For any point in the plane of the
triangle, the same representation works but the b
i
may be any real numbers. Given
X, you may solve for the barycentric coordinates:
X − P
0
= (b
0
− 1)P
0
+ b
1
P
1
+ b
2
P
2
= b
1
(P
1
− P
0
) + b
2
(P
2
− P
0
)
Applying various dot products, you may set up the linear system of equations
(P
1
− P
0
)
.
(P
1
− P
0
)(P
1
− P
0
)
.
(P
2
− P
0
)
(P
2
− P
0
)
.
(P
1
− P
0
)(P
2
− P
0
)
.
(P
2
− P
0
)
b
1
b
2
=
(P
1
− P
0
)
.
(X − P
0
)
(