dorsal/arxiv
View SchemaGeneric Smooth Connection Functions - A New Analytic Approach to Interpolation
| Authors | Alex Alon, Sven Bergmann |
|---|---|
| Categories | |
| ArXiv ID | physics/0105039 |
| URL | https://arxiv.org/abs/physics/0105039 |
| DOI | 10.1088/0305-4470/35/17/305 |
Abstract
We present a generic solution to the fundamental problem of how to connect two points in a plane by a smooth curve that goes through these points with a given slope. The smoothness of any curve depends both on its curvature and its length. The smoothest curves correspond to a particular compromise between minimal curvature and minimal length. They can be described by a class of functions that satisfy certain boundary conditions and minimize a weight functional. The value of this functional is given essentially by the average of the curvature raised to some power \nu times the length of the curve. The parameter \nu determines the importance of minimal curvature with respect to minimal length. In order to find the functions that obtain the minimal weight, we use extensively notions that are well-known in classical mechanics. The minimization of the weight functional via the Euler-Lagrange formalism leads to a highly non-trivial differential equation. Using the symmetries of the problem it is possible to find conserved quantities, that help to simplify the problem to a level where the solution functions can be written in a closed form for any given \nu. Applying the appropriate coordinate transformation to these solutions allows to adjust them to all possible boundary conditions.
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"abstract": "We present a generic solution to the fundamental problem of how to connect\ntwo points in a plane by a smooth curve that goes through these points with a\ngiven slope. The smoothness of any curve depends both on its curvature and its\nlength. The smoothest curves correspond to a particular compromise between\nminimal curvature and minimal length. They can be described by a class of\nfunctions that satisfy certain boundary conditions and minimize a weight\nfunctional. The value of this functional is given essentially by the average of\nthe curvature raised to some power \\nu times the length of the curve. The\nparameter \\nu determines the importance of minimal curvature with respect to\nminimal length. In order to find the functions that obtain the minimal weight,\nwe use extensively notions that are well-known in classical mechanics. The\nminimization of the weight functional via the Euler-Lagrange formalism leads to\na highly non-trivial differential equation. Using the symmetries of the problem\nit is possible to find conserved quantities, that help to simplify the problem\nto a level where the solution functions can be written in a closed form for any\ngiven \\nu. Applying the appropriate coordinate transformation to these\nsolutions allows to adjust them to all possible boundary conditions.",
"arxiv_id": "physics/0105039",
"authors": [
"Alex Alon",
"Sven Bergmann"
],
"categories": [
"physics.class-ph",
"physics.ed-ph",
"physics.gen-ph"
],
"doi": "10.1088/0305-4470/35/17/305",
"title": "Generic Smooth Connection Functions - A New Analytic Approach to Interpolation",
"url": "https://arxiv.org/abs/physics/0105039"
},
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