OSCULATING MIRROR. An osculating plane is the plane in which a curve sits. Simply put, the intersection of a plane with a developable surface defines a planar curve, and that curve is a potential ridge for the surface.
GEODESIC DUPLICATION. By mapping a geodesic curve (a straight line in the plane development) onto a 3-dimensional surface, a space ridge connecting identical folia of a single surface is defined.
CONIC CUSP. In resolving terminating ridges, the point of their convergence can be conceived as a cusp, which serves as the focus for the generators of a general conic surface. This negotiating surface lies tangent to each of the adjacent developables.
PARABOLIC TRANSFORMATION. It is possible to graft into a single surface two kinds of developability: conic and cylindrical. This requires that parallel generators of the cylindrical surface be reflected across normals to a parabolic curve in the plane development, converging at the focus of the parabola. This point, mapped to 3-dimensional space, is the cusp of a cone whose base is the parabola.
There exists a longstanding discourse on developable surfaces in architecture and elsewhere. The key characteristic of developable surfaces, that which makes them of interest in many fields outside geometry, is their isometry in plane development. The short-hand description of this characteristic is to say they “unroll” onto a flat plane without distortion (stretching or tearing). Commonly called “paper shapes,” developable surfaces are governed by the same constraints that govern the behavior of most sheet materials.
The current discourse around developability is limited to “smooth” developable shapes. Such surfaces may have trimmed edges, and at most an edge of regression. In short, they are parameterized in a narrow way. An expanded and more productive view includes folds (a.k.a. catastrophes) within the parameterized neighborhood.
The figured catastrophe (a ridge of non-zero curvature) is the general case of all folds, and therefore is the subject of this research.
The fundamental contribution of this research lies in the creation of four graphical theorems. As in mathematics, these theorems consolidate related ideas into a useful building block for further investigation and discovery.
In each case, a geometric/graphic method is chosen, despite the availability of algebraic, parametric or algorithmic equivalents. This is a designer’s bias that distinguishes this effort from geometry aimed at mathematicians. The explicit aim here is instead to expand the set of possibilities available to designers.