Barycentric Coordinates for Arbitrary Polygons in the Plane

22 K. Hormann
twice) and also in the case that Ψ is a set of (possibly nested) polygons, we conclude
that
n∑
κ i > 0.
i=1
But the sum of the κ i can be rearranged, by a change of summation index, to be half the
sum of the w i (v) in the form of Equation (11) and therefore W (v) is positive for any v
inside Ψ. Likewise one can show that W (v) is negative over the exterior of Ψ.
C
Lagrange and Linearity Property
From the representation of the homogeneous coordinates in Equation (11) it follows
that w i (v) is C ∞ everywhere except at the vertices v i−1 , v i , v i+1 and the edges e i−1 ,
e i . As a consequence, the normalized coordinates λ i (v) are C ∞ at all v /∈ Ψ. To study
the behaviour of λ i on one of the edges e j of Ψ, we consider the slightly modified
homogeneous coordinates
ŵ i (v) = w i (v)A j (v) = r j(v)r j+1 (v)
(z i−1 (v) + z i (v)),
r i (v)
where z i (v) = tan(α i (v)/2) sin(α j (v))/2. Then we have for any v ∈ e j that z i (v) = 0
for i ≠ j and z j (v) = 1 and therefore ŵ j (v) = r j+1 (v), ŵ j+1 (v) = r j (v), and
ŵ i (v) = 0 for i ≠ j, j + 1. It follows that λ i (v) = 0 for all i ≠ j, j + 1 and
λ j (v) =
r j+1 (v)
r j (v) + r j+1 (v) , λ r j (v)
j+1(v) =
r j (v) + r j+1 (v) ,
which are the standard barycentric coordinates on an edge, converging to δ ij as v approaches
v j . Further it is clear that the functions ŵ i are C ∞ over e j and so are the λ i .
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