Normal variation for adaptive feature size

Claim 1 Let q and q′ be any two points in Σ so that d(q, q′) ≤ εmin{f(q), f(q′)} for ε ≤ 1 3 . Then, ∠nq, nq′ ≤ ε 1−3ε . Unfortunately, the proof of this claim as given in Amenta and Bern [1] is wrong; it also appears in the book by Dey [2]. In this short note, we provide a correct proof with an improved bound of ε 1−ε . Theorem 2 Let q and q′ be two points in Σ with d(q, q′) ≤ εf(q) where ε ≤ 13 . Then, ∠nq, nq′ ≤ ε 1−ε .