Journal
RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS
Volume 20, Issue 3, Pages 257-267Publisher
PLEIADES PUBLISHING INC
DOI: 10.1134/S1061920813030011
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Funding
- RFBR-CNRS-a [10-01-93111]
- RFBR [12-01-00748-a]
- Austrian Science Fund (FWF) [M 1273-N18]
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In this paper, we study normal forms of plane curves and knots. We investigate the Euler functional E (the integral of the square of the curvature along the given curve) for closed plane curves, and introduce a closely related functional A, defined for polygonal curves in the plane ae(2) and its modified version A (R) , defined for polygonal knots in Euclidean space ae(3). For closed plane curves, we find the critical points of E and, among them, distinguish the minima of E, which give us the normal forms of plane curves. The minimization of the functional A for plane curves, implemented in a computer animation, gives a very visual approximation of the process of gradient descent along the Euler functional E and, thereby, illustrates the homotopy in the proof of the classical Whitney-Graustein theorem. In ae(3), the minimization of A (R) (implemented in a 3D animation) shows how classical knots (or more precisely thin knotted solid tori, which model resilient closed wire curves in space) are isotoped to normal forms.
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