On the usage of the curvature for the comparison of planar curves

Given two curves, on the plane or in space, or surfaces, looking for a deformation from one into another such that the transformation is gradual and continuous is an open and interesting problem both from a theoretical stand point and for the applications that can be envisaged. In particular, if the two starting shapes have some particular features, we expect that also the intermediate ones preserve the same characteristics. This thesis focuses on planar closed curves, aiming to find deformations between two planar closed curves, preserving the closeness. Our approach is treating curves in the perspective of Differential Geometry, that is represented by parameterizations. We show how the simple idea of linearly interpolating between the two parameterizations is too much dependent on the mutual position of the curves, implicit in the parameterizations. This suggested us to consider a deformation based on intrinsic properties of curves, in particular we took in account the curvature. The fundamental theorem of planar curves states that, given the signed curvature function with respect to arc length, it is possible to reconstruct the curve from it up to rigid motions which preserve the orientation. Motivated by these observations we linearly interpolate the curvature of source and target curves, parameterized with respect to the arc length, and we reconstruct the corresponding intermediate curves. Unfortunately, the curvature interpolation not always leads to closed curves. We overwhelmed this limitations replacing each intermediate open curve with the closed curve as close as possible to the open one measuring the same length. To do this we need to define an appropriate distance between curves. A distance based on the mutual position of the curves and dependent on their particular parameterizations is not feasible for our purposes, so as measure of distance between two curves we consider the distance between their curvatures. This paradigm shift leads to find the curve with the curvature as close as possible to the curvature of the open, interpolated, one. This decision is supported by the proof that there is a link between close curvatures and close curves, or to be more precise, that the distance between curvatures gives a bound for the distance between corresponding curves with a particular mutual position, where distances are computed with appropriate metric. We also show that solving this problem in the smooth setting is very difficult since it is not a classical variational problem, so we propose a simple example where we try to solve the variational problem in the smooth setting and then we conclude giving an approximated solution for the general case.