Largest Placement of One Convex Polygon Inside Another

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 logn time We also show that the combinatorial complexity of the space of all similar copies of P inside Q is O mn and that it can also be computed in O mn logn time Let P be a convex polygon with m edges and Q a convex polygon with n edges Our goal is to nd the largest similar copy of P inside Q allowing translation rotation and scaling of P see Figure A restricted version of this problem in which we just determine whether P can be placed inside Q without scaling was solved by Chazelle in O mn time See also 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 n in O m n in and O m n in 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 based on matching two dimensional faces of polyhedral objects The transformation which places the largest similar copy of a polygon P derived from a face of an object model inside a polygon Q derived from an image is a candidate for a transformation which matches the entire model to the image Pankaj Agarwal and Micha Sharir have been supported by a grant from the U S Israeli Binational Science Foundation Pankaj Agarwal has also been supported by National Science Foundation Grant CCR an NYI award and by matching funds from Xerox Corp Nina Amenta was employed by the Geometry Center which is o cially the NSF Center for Computation and Visualization of Geometric Struc tures supported by NSF DMS Micha Sharir has also been supported by NSF Grants CCR and CCR by a Max Planck Research Award and by grants from the Israel Science Fund administered by the Israeli Academy of Sciences and the G I F the German Israeli Foundation for Scienti c Research and Development Department of Computer Science Box Duke University Durham NC USA Xerox PARC Coyote Hill Road Palo Alto CA USA School of Mathematical Sciences Tel Aviv University Tel Aviv ISRAEL and Courant Institute of Mathematical Sciences New York University New York NY USA