first quadratic forms may be written as follows The condition of conformal mapping for two surfaces as is formulated in differential geometry requires that where is the ratio of the surface distortion due to conformal mapping. It is known that the first quadratic form reflects the distance between two surface points and . When -ratio is close to 1 the above eqn converges to condition of isometric mapping and to equi-areal mapping respectively because of invariant angles between any curves and invariance of any pieces of area. Remembering that the first stage of form finding is based on triangular mesh of a surface and using the method of weighted residuals for the description of isometric and equi-areal mapping of the minimum surface onto a plane area we may write the following function which is defined by the sum of integrals along segments of curved triangles where - total number of grid cells, - weight ratios, - the total mapping residual, - the constant that does not influence the final result and may be used as a scale ratio. Considering further weight ratios we may transform eqn. into approximate finite sum that is a combination of linear distances between nodes of the surface grid and write the basic condition of equi-areal surface mapping as a minimum of following non-linear function where - initial length of linear segment number , - final length of segment number , - distortion ratio close to 1 and may be different for each segment. The initial and final lengths of segment number may be expressed as usual by two nodal co-ordinates as where - co-ordinates of nodes of the initial segment, - co-ordinates of nodes of the final segment. According to the initial assumption we can write for the plane surface mapping. The expression for vectors and with co-ordinate increments term use may be written as The vector definition is made as previously After transformations we may write the following two independent systems of non-linear algebraic equations where all the parts of the system can be expressed as previously and and are vectors of pseudo-stresses at axes 1, 2 that has the following form where - total number of nodes that surround node number , - the number of global axes. The above approach is another form of SGM and allows the obtaining of two independent systems of non-linear algebraic equations that can be solved by