Fitting Polynomial Surfaces to Triangular Meshes with Voronoi ...

16 Vincent Nivoliers, Dong-Ming Yan, and Bruno Lévy
Fig. 14: Configuration of the nearest point of Π S (y).
The bound follows from the fact that B ε ̸⊂ B Vor ∪ B l f s . Let p be a point of
B Vor ∩ B l f s . This point exists since ε < 2 and B ε is not included in B Vor .
Using this point the previous condition can be reformulated as p ∈ B ε .
Using triangular identities in (Π S (y), c l f s , p), we have:
‖ Π S (y) − p ‖ 2 = 2lfs(Π S (y)) 2 (1 − cos α) (8)
with α being the (p, c l f s , Π S (y)) angle. Using the same identities in (x, c l f s , p)
we obtain:
˜ d 2 = (d + lfs(Π S (y))) 2 + lfs(Π S (y)) 2
− 2(d + lfs(Π S (y)))lfs(Π S (y)) cos α
d˜
2 − d 2 = 2(d + lfs(Π S (y)))lfs(Π S (y))(1 − cos α)
Using equation 8, (1 − cos α) can be replaced:
d˜
2 − d 2 = (d + lfs(Π S (y))) ‖ Π S(y) − p ‖ 2
lfs(Π S (y))
(9)
Finally since p is inside B ε , we have:
This finally provides the result:
‖ Π S (y) − p ‖≤ εlfs(Π S (y)) (10)
˜ d 2 − d 2 ≤ ε 2 lfs(Π S (y))(lfs(Π S (y)) + d) (11)
This bound is sharp since it is reached whenever S is exactly B l f s and x i is
located at p.