Euler elasticae in the plane and the Whitney-Graustein theorem
Tóm tắt
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 ℝ2 and its modified version A
R
, defined for polygonal knots in Euclidean space ℝ3. 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 ℝ3, 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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