 
Summary: Normal variation for adaptive feature size
Nina Amenta
and Tamal K. Dey
September 17, 2007
Background
Let be a closed, smooth surface in R3. For any two sets X, Y R3, let d(X, Y )
denote the Euclidean distance between X and Y . The local feature size f(x) at a
point x is defined to be the distance d(x, M) where M is the medial axis of
. Let np denote the unit normal (inward) to at point p. Amenta and Bern in
their paper [1] claimed the following:
Claim 1 Let q and q be any two points in so that d(q, q ) min{f(q), f(q )}
for 1
3 . Then, nq, nq
13 .
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
1 .
Theorem 2 Let q and q be two points in with d(q, q ) f(q) where 1
3 .
