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Minima and Saddles in the MLS Surface Definition Nina Amenta

Summary: Minima and Saddles in the MLS Surface Definition
Nina Amenta
July 9, 2008
Shlomo Gortler and Zachary Abel (a Harvard undergrad) point out (personal communication) a
problem in the proof of Claim 1 in our paper, "Defining Point-Set Surfaces" [1]. The Claim concerns
the MLS surface [2], which is defined in terms of the MLS energy function eMLS(y, a), a function on
the space IR3
P2. We first define the set Jx of point-direction pairs {(y, a) a = (y-x)/len(y-x)},
that is, a is the unit direction vector from x to y. Then Levin's definition can be stated:
Definition: A point x belongs to the MLS surface if and only if x is a local minimum of eMLS(y, a)
restricted to the set Jx.
Our Claim was:
Claim 1 The MLS surface consists of the points for which n(x) is well-defined, and for which
x arglocalminyLx,n(x)
eMLS(y, n(x)) (1)
Here x is a point in IR3
, n(x) is the unit direction vector minimizing eMLS(x, a) over all a, and
Lx,n(x) is the line through x with direction n(x).
This Claim is an if-and-only-if statement; unfortunately only one direction is true. A point of
the MLS surface does indeed satisfy Equation (1), but as Shlomo and Zachary point out there may


Source: Amenta, Nina - Department of Computer Science, University of California, Davis


Collections: Biology and Medicine; Computer Technologies and Information Sciences