The boundary of a shape and its classification

A unified algebraic foundation for shape computation is presented wherein the description of a shape is explored as a sum of disjoint segments and theresult of shape computation is expressed in terms of a classification of the boundaries of these segments. Shapes are considered as collections of spatial elements of limited but nonzero measure, independently of dimensionality or shape type.  A spatial element is itself specified by two shapes: a carrier and a boundary.  The carrier is a shape in which the element is embedded and is of the same type as the element.  The boundary represents the form of the element and is a shape of a different type.  A particular kind of spatial element or shape is a segment that has no nonempty proper subshape, the boundary of which is a subshape of the boundary of the segment.  It is shown that a shape is the sum of a unique finite set of disjoint segments with disjoint boundaries.  Then, the shape is said to be maximal and the boundary of a maximal shape is the sum of the boundaries of its maximal segments. Boundary segments of a shape can be classified with respect to another shape as to be inside or outside the other shape, or shared in the same way or shared oppositely between the two shapes.  From this classification, the boundary of a shape resulting from a shape operation on two shapes is determined by summing appropriate classes of segments.  In a similar way, shape relations between two shapes are shown to depend on the distribution of the boundary segments of each shape into these classes.