This chapter contains all necessary information in regard to the differential 
geometry of curves and surfaces needed for the description of membrane shapes. 
The main focus is on the geometry of the plane curves to which most of the 
considerations in the book boil down. As usual, coordinate systems, curvature 
and the Frenet-Serret equations are discussed in some detail. Then, space curves 
together with their tangent vectors, normal planes and curvature are introduced. 
Other important notions like principal normal and binormal, osculating plane and 
moving frame are introduced, exemplified and discussed as well. The Frenet-
Serret equations for the space curves are derived and the principal theorem in 
the local theory of these curves is proved. The second part of this chapter is 
devoted to variational calculus, as this is the setting in which the problem of 
the optimal form of membranes is cast later on. Variational calculus occupies a 
special place in mathematics. Many of the laws of nature are formulated as 
variational principals. Hence, there is a great interest in the calculus of 
variations in many applied areas—from celestial mechanics to mathematical 
economics and management theory. Specifically in this chapter, the equations 
named after Euler and Lagrange are reviewed (and in some cases derived) in 
various settings, e.g., for functionals depending on one or more variables and 
comprising derivatives of first or higher order, etc. Most of these cases are 
illustrated by examples, which serve both to assist in understanding and to 
prepare the reader for the applications of the method to follow.