Affine subspaces of curvature functions from closed planar curves
Abstract
Given a pair of real functions (k,f), we study the conditions they must satisfy for k+λ f to be the curvature in the arc-length of a closed planar curve for all real λ. Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied. Finally, the characterization obtained is used to show that a sufficient analogue of the 4-vertex theorem cannot be developed.
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