Optimal mappings onto a plane

The work described in the thesis is directly related to footwear technology, and may have wider applications. The upper and insole of a shoe are made from components cut from flat sheet material, which must be shaped to conform to the curved surface of a last. The pattern for the component is a map of the curved surface. Two methods for constructing plane maps of curved surfaces are described. One is based on a tree drawn on the surface, whose branches are each mapped isometrically on to a plane. Some well-known map projections used by cartographers fall into this category, as does a patented method of constructing shoe patterns. The second is an optimisation method: the ideal mapping is an isometry, (only possible if the curved surface is developable) and a norm is constructed which measures the amount by which any actual mapping departs from an isometry. The optimal mapping is the one which minimises the norm. Analysis shows that the principal directions at the boundary of the map are along and at right angles to the boundary, and that the principal scale across the boundary is unity, but the actual construction of an optimal map can only be performed by an iterative computer program. Examples are given of maps of shoe components and of portions of a sphere; the maps may include plots of the principal lines. The limitations imposed by the presence of seams are indicated, and a theory of the closed seam is worked out. The whole idea of optimal maps is set in the context of a comprehensive programme for the computer-aided design of footwear.