What shape grammars do that CAD should: the 14 cases of shape embedding

Abstract Shape queries based on shape embedding under a given Euclidean, affine, or linear transformation are absent from current CAD systems. The only systems that have attempted to implement shape embedding are the shape grammar interpreters albeit with promising but inconclusive results. The work here identifies all possible 14 cases of shape embedding with respect to the number of available registration points, four for determinate cases and ten for indeterminate ones, and an approach is sketched to take on the complexities underlying the indeterminate cases. All visual calculations are done with shapes consisting of straight lines in the Euclidean plane within the algebra Uij for i = 1 the dimension of lines and j = 2 the dimension of space in which the lines are defined, transformed and combined. Aspects of interface design and integration to current work design workflows are deliberately left aside.