While principal curvature lines seem to be currently very popular, they’ve been around in architecture since 1796.

Gaspard Monge, a french geometer, proposed the above design for an ellipsoidal vault that could be constructed out of mortarless masonry units. Not only did he come up with a completely new way to build such vaults, he actually solved a much bigger problem: how to construct freeform surfaces out of discrete units (see [1], [2], and [3]).

One major characteristic of the patterns that the principal curvature lines of a surface create is that while they are stable for small disturbances of the surface, larger disturbances will cause the overall pattern to “jump” to a new one. This kind of behaviour is called a bifurcation. Now, designing something that wants to bifurcate every time it is modified is fairly difficult. Maybe an understanding of what kinds of surfaces have the same principal curvature line pattern would make this easier? Limiting the design to such surfaces would embed the rationality of the curvature line pattern directly into the design process.