 
Summary: Largest Placement of One Convex Polygon inside Another \Lambda
Pankaj K. Agarwal y Nina Amenta z Micha Sharir x
December 5, 1995
Abstract
We show that the largest similar copy of a convex polygon P with m edges inside a
convex polygon Q with n edges can be computed in O(mn 2 log n) time. We also show
that the combinatorial complexity of the space of all similar copies of P inside Q is
O(mn 2 ), and that it can also be computed in O(mn 2 log n) time.
Let P be a convex polygon with m edges and Q a convex polygon with n edges. Our
goal is to find the largest similar copy of P inside Q (allowing translation, rotation, and
scaling of P ); see Figure 1. A restricted version of this problem, in which we just determine
whether P can be placed inside Q without scaling, was solved by Chazelle [4], in O(mn 2 )
time. See also [1, 6, 12] for other approaches to the more general problem, in which Q is
an arbitrary polygonal region. (We remark that the complexity of the algorithms for the
general case is considerably higher, about O(m 2 n 2 ) in [1], O(m 3 n 2 ) in [12], and O(m 4 n 2 )
in [6].)
Problems concerning the placement of one polygon inside another are important in
robotics and manufacturing. This restricted problem is also applicable to an approach
to object recognition recently proposed by Basri and Jacobs [3], based on matching two
dimensional faces of polyhedral objects. The transformation which places the largest similar
