Computing the center of area of a polygon
Joseph O'Rourke1989, Lecture Notes in Computer Science
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The center of area of a polygon P is the unique point p* that maximizes the minimum area overlap between P and any halfplane that includes p*. We present several "numerical" algorithms for finding the coordinates of p* for a polygon of n vertices. These algorithms are numerical in the sense that we have been careful to
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Computing the Center of Area of a Polygon
Matthew Diaz
Department of Computer Science
The Johns Hopkins University
Baltimore, MD 21218
Joseph O'Rourke
Department of Computer Science
Smith College
Northampton, MA 01063
Abstract
The center of area of a polygon P is the unique point p* that maximizes the
minimum area overlap between P and any halfplane that includes p*. We present
several "numerical" algorithms for finding the coordinates of p* for a polygon of n
vertices. These algorithms are numerical in the sense that we have been careful to
express the algorithm complexities as a function of G, the number of bits used to
represent the coordinates of the polygon vertices, and K, the number of desired bits
of precision in the output coordinates of p*. For a convex polygon the algorithm runs
in O(nGK); non-convex polygons offer considerably more challenge. For orthogonal
non-convex polygons, we have an algorithm that runs in O(n2GK), but for general
non-convex polygons, our algorithm's time complexity is O(n 4 log nGK + nSG2K +
nG K2).
1
Introduction
Given a set of points, Q, in the plane, one may wish to find a point p (not necessarily in
Q), such that the minimum number of points contained in any halfplane that includes p
is maximized. This point is called a centerpoint, and algorithms for computing it and the
related k-hulls are known [Ede87, p. 63], [CSY87].
The natural generalization of this discrete question to the continuous domain is the
following: Given a polygon P, which point p has the property that the minimum area
overlap of any halfplane which includes p is maximized? This (unique) point we call the
center of area.
Although we examine traditional combinatorial computational geometry Mgorithms
for finding the center of area, it seems clear that when the output of an algorithm is a
single point not directly present in the input, combinatorial complexity should not be
the sole measure of worth: the cost per bit of output in the point's coordinates is the
172
fundamental issue. Thus we take care to deal with numerical precision, which leads us to
search algorithms rather different from the combinatorial algorithms.
Section 2 introduces the notation that will be used through this paper and discusses
certain geometric properties of the center of area and related structures. Section 3 introduces a necessary condition for a halfplane to minimize the area overlap, and Section 4
discusses a combinatorial structure, the chord diagram, associated with the center of area.
An algorithm for finding the center of area is presented in Section 5, including analysis for
convex, orthogonal and general polygons. We conclude in Section 6 with generalizations
and open problems. Due to space limitations many of the proofs are limited to sketches
only. For a more complete analysis refer to [DO88] and [DO89].
2
Definitions
Define L(p, O) to be a directed line through the point p forming an angle 0 with the
positive x axis, and let L+(p, O) be the closed halfplane bounded by, and to the left of
L(p, 0), and L-(p, O) be the closed halfplane bounded by, and to the right of L(p, 0).
Hence L(p, O) = L+(p, 8) n L-(p, 8).
For a polygon P, define C(p, O) to be the set of collinear chords formed by the intersection of L(p,O) with P. We then define the function a(p,O) = ]L+(p,O) n P[, the area
of P to the left of the chords C(p,O). (Figure 1)
P
p
p,8)
Figure h Notation
Define
a(p) = min a(p, 0).
O
Hence a(p) is the minimum area cut off by any C(p, 0). We will let O* represent any one of
the angles that minimizes a(p). (It is possible for the minimum to be achieved at several
distinct angles.) We will refer to L(p,O*) and C(p,O*) as a min-line and min-chordset
respectively.
The center of area for a simple polygon is defined as the set of points p* that satisfy
a(p* ) = max
a(p).
p
The value a(p*) we denote by 6% For convex polygons the ratio of the area of L+(p*, O*)NP
to the area of L-(p*, 0") n P is known as Winternitz's Measure of Symmetry [Gru63].