Barycentric Coordinates for Arbitrary Polygons in the Plane

20 K. Hormann
v j+2
v
Ψ^
r j+1
α j+1
^
α α +
r^
j ^
α j–1
α –
r j
v j–1
v^
v j
v j+1
Figure 13: Notation for refining a polygon by adding a vertex.
A
Refinability
If we refine the polygon Ψ to ̂Ψ by adding a vertex ˆv between v j and v j+1 as in Figure
13, then it follows from the locality of the homogeneous coordinates that ŵ i = w i
for i ≠ j, j + 1. According to Equation (11), the other three homogeneous coordinates
are
ŵ j (v) = 2 ( tan(α j−1 (v)/2) + tan(̂α − (v)/2) ) /r j (v),
ŵ(v) = 2 ( tan(̂α − (v)/2) + tan(̂α + (v)/2) ) /ˆr(v),
ŵ j (v) = 2 ( tan(̂α + (v)/2) + tan(α j+1 (v)/2) ) /r j+1 (v).
If we now write ˆv as an affine combination of v, v j , and v j+1 , i.e.,
with barycentric coordinates
and
ˆv = ρ(v)v + σ(v)v j + τ(v)v j+1
σ(v) = ˆr(v) sin ̂α+ (v)
r j (v) sin α j (v) , τ(v) = ˆr(v) sin ̂α− (v)
r j+1 (v) sin α j (v) ,
ρ(v) = 1 − σ(v) − τ(v),
then some elementary transformations show that the homogeneous coordinates w j and
w j+1 can be expressed as
w j (v) = ŵ j (v) + σ(v)ŵ(v),
w j+1 (v) = ŵ j+1 (v) + τ(v)ŵ(v),
and that the sum of the refined homogeneous coordinates satisfies
Ŵ (v) = W (v) + ρ(v)ŵ(v).
It further follows that Ŵ (v) = W (v) in the special case that ˆv lies on the edge e j =
(v j , v j+1 ) since then ρ(v) = 0.
INSTITUT FÜR INFORMATIK