GRAPHO-ANALYTICAL APPROACH FOR OBTAINING THE UNFOLDINGS OF NON–EXTENDABLE SURFACES OF THE SECOND AND HIGHER DEGREES OF DETAILS IN THE WOODWORKING AND FURNITURE INDUSTRY

Abstract

This article presents graphical–analytical methods for approximating the developments of non-developable surfaces for components used in the woodworking and furniture industries. Surface development is a key design stage that converts a three-dimensional object into a two-dimen-sional cutting pattern. Double-curvature surfaces, such as spheres, ellipsoids, and paraboloids, cannot be developed onto a plane without distortion, tearing, or overlap. They must therefore be approximated by segments of developable cylindrical or conical surfaces. The increasing demand for graphical accuracy and technological precision makes this problem particularly relevant. The aim of the study is to obtain approximate developments of toroidal, spherical, and ellipsoidal surfaces by combining graphical construction with analytical calculations. The method replaces a theoretically non-developable surface with a set of small developable seg-ments that can be unfolded with minimal distortion. This decomposition enables complex shapes to be manufactured from sheet material using conventional rolling or bending processes. Analytical calculations confirm that errors are unavoidable in purely graphical developments because of the approximation itself. Computer-aided design (CAD) reduces these errors by allowing individual surface segments to be constructed at a larger scale rather than drawing the entire complex geometry at once. The approach is applicable to the design and manufacture of industrial tanks, silos, sheet-metal dome roofs, spiral staircases, decorative columns, and transport augers.