Within the framework of the well-known curvature models, a fluid lipid bilayer membrane is regarded as a surface embedded in the three-dimensional Euclidean space whose equilibrium shapes are described in terms of its mean and Gaussian curvatures by the so-called membrane shape equation. In the present paper, all solutions to this equation determining cylindrical membrane shapes are found and presented, together with the expressions for the corresponding position vectors, in explicit analytic form. The necessary and sufficient conditions for such a surface to be closed are derived and several sufficient conditions for its directrix to be simple or self-intersecting are given.

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Here we derive analytic expressions for the scalar parameters which appear in the generalized Euler decomposition of the rotational matrices in R 3. The axes of rotations in the decomposition are almost arbitrary and they need only to obey a simple condition to guarantee that the problem is well posed. A special attention is given to the case when the rotation is decomposable using only two rotations and for this case quite elegant expressions for the parameters were derived. In certain cases one encounters infinite parameters due to the rotations by an angle π (the so called half turns). We utilize both geometric and algebraic methods to obtain those conditions that can be used to predict and deal with various configurations of that kind and then, applying l'Hôpital's rule, we easily obtain the solutions in terms of linear fractional functions. The results are summarized in two Tables and a flowchart presenting in full details the procedure. Contents 1 Introduction 60 2 The Generic Case 64 3 The Symmetric Case 69 4 Decomposition Into Two Rotations 71

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The consideration of some non-standard parametric Lagrangian leads to a fictitious dynamical system which turns out to be equivalent to the Euler problem for finding out all possible shapes of the lamina. Integrating the respective differential equations one arrives at novel explicit parameterizations of the Euler's elastica curves. The geometry of the inflexional elastica and especially that of the figure "eight" shape is studied in some detail and the close relationship between the elastica problem and mathematical pendulum is outlined.