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
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.
A doctoral thesis submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy at Loughborough University.
Great Britain, Department of Industry, Garment and Allied Industries Requirements Board (GARB).