| | |
Summary: PSEUDO-TRIANGULATIONS FROM SURFACES AND A NOVEL
TYPE OF EDGE FLIP
OSWIN AICHHOLZER , FRANZ AURENHAMMER, HANNES KRASSER y , AND
PETER BRASS z
Abstract. We prove that planar pseudo-triangulations have realizations as polyhedral surfaces
in three-space. Two main implications are presented: The spatial embedding leads to a novel
ip
operation that allows for a drastical reduction of
ip distances, especially between (full) triangu-
lations. Moreover, several key results for triangulations, like
ipping to optimality, (constrained)
Delaunayhood, and a convex polytope representation, are extended to pseudo-triangulations in a
natural way.
Key words. Pseudo-triangulation,
ip distance, surface realization, locally convex function,
constrained regular complex, polytope representation
AMS subject classi cations. 68U05, 68W40, 52C45, 52C25, 52B55, 52B05
1. Introduction. In geometric data processing, structures that partition the
geometric input, as well as connectivity structures for geometric objects, play an
important role. A versatile tool in this context are triangular meshes, often called
triangulations; see e.g., the survey articles [14, 26, 11]. A triangulation of a nite
set S of points in the plane is a maximal planar straight-line graph that uses exactly
the points in S as its vertices. Each face in a triangulation is a triangle spanned by S.
In recent years, a relaxation of triangulations, called pseudo-triangulation (or
|
