1. INTRODUCTION. Let r be a finite family of simple curves in the plane. When is there a homeomorphism of the plane to itself that takes all the curves in r to straight lines? In the Euclidean plane, E2, we are faced with the fact that two non-intersecting curves in our family must map to two parallel lines. This introduces extraneous technical complications that only distract from the essence of the problem. As with many other geometric questions, it is much simpler to avoid the special cases caused by parallel lines by moving to the projective plane. The real projective plane p2 iS the Euclidean plane E2 with an extra "line at infinity" adjoined, each point of which represents a parallel direction in E2. p2 has the virtue of simplicity: every pair of points determines a unique line which is topologically a circle (i.e., a simple closed curve), and every two lines meet at a unique point. Thus our question becomes: When is there a homeomorphism of p2 to itself that simultaneously straightens all the members of a finite family r of simple closed curves?
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