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We study the existence of the two affine-equivalent bar-and-joint frameworks in Euclidean -space which have some prescribed combinatorial structure and edge lengths. We show that the existence problem is always solvable theoretically and explain why to propose a practical algorithm for solving the problem is impossible. [ABSTRACT FROM AUTHOR]

Using the Green’s theorem we reduce the variation of the total mean curvature of a smooth surface in the Euclidean 3-space to a line integral of a special vector field, which immediately yields the following well-known theorem: the total mean curvature of a closed smooth surface in the Euclidean 3-space is stationary under an infinitesimal flex. [ABSTRACT FROM AUTHOR]