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==Generalizations==
The curves of constant width can be generalized to certain non-convex curves, the curves that have two tangent lines in each direction, with the same separation between these two lines regardless of their direction. As a limiting case, the [[Hedgehog (geometry)|projective hedgehogs]] (curves with one tangent line in each direction) have also been called "curves of zero width".{{r|kelly}}
One way to generalize these concepts to three dimensions is through the [[surface of constant width|surfaces of constant width]]. The three-dimensional analog of a Reuleaux triangle, the [[Reuleaux tetrahedron]], does not have constant width, but minor changes to it produce the [[Meissner bodies]], which do.{{r|gardner|mmo}} The curves of constant width may also be generalized to the [[Body of constant brightness|bodies of constant brightness]], three-dimensional shapes whose two-dimensional projections all have equal area; these shapes obey a generalization of Barbier's theorem.{{r|mmo}} A different class of three-dimensional generalizations, the [[space curve]]s of constant width, are defined by the properties that each plane that crosses the curve perpendicularly intersects it at exactly one other point, where it is also perpendicular, and that all pairs of points intersected by perpendicular planes are the same distance apart.{{r|fujiwara|cieslak|teufel|wegner72}}
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