On the plane curves whose curvature depends on the distance from the origin. (English) Zbl 1260.53012

Mladenov, I. M.; Hadzhilazova, M. Ts.; Djondjorov, P. A.; Vassllev, V. M.

Kielanowski, Piotr (ed.) et al., XXIX workshop on geometric methods in physics, Białowieża, Poland, June 27 – July 3, 2010. Selected papers based on the presentations at the workshop. Melville, NY: American Institute of Physics (AIP) (ISBN 978-0-7354-0861-6/hbk). AIP Conference Proceedings 1307, 112-118 (2010).

Summary: Here we suggest and exemplify a simple scheme for reconstruction of a plane curve if its curvature belonging to the class specified in the title by deriving explicit parametrizations of Bernoulli’s lemniscate and newly introduced co-lemniscate curve in terms of the Jacobian elliptic functions. The relation between them and the Bernoulli elastica are clarified.
For the entire collection see [Zbl 1234.81018].

MSC:

53A04	Curves in Euclidean and related spaces
74K05	Strings

Keywords:

classical differential geometry; plane curves; curvature; elasticity theory