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We describe the parameter spaces of some families of quadrilaterals, such as parallelograms, rectangles, rhombuses, cyclic quadrilaterals and trapezoids. For this purpose, we prove that the closed
is the unique topological
-manifold (with boundary) whose boundary and interior are homeomorphic to
, respectively. Roughly speaking, our main result states that the natural compactifications of the parameter spaces of cyclic quadrilaterals and of trapezoids, modulo similarity, are both homeomorphic to
The space of shapes of quadrilaterals can be identified with
. We deal with the subset of
corresponding to convex quadrilaterals and the subset which corresponds to simple (that is, without self-intersections) quadrilaterals. We provide a complete description of the topological closures in
of both spaces. Although the interior of each space is homeomorphic to a disjoint union
, their closures are topologically different. In particular, the boundary of the space corresponding to convex quadrilaterals is homeomorphic to a pair of three-dimensional spheres glued along a Möbius strip while the boundary of the space corresponding to simple quadrilaterals is more complicated.
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