Design and Optimization of Orthogonally Intersecting Planar Surfaces

We present a method for the design of 3D constructions from planar pieces that can be cut easily and cheaply with laser cutters and similar Computer Numerical Control (CNC) machines. By cutting tight slits in intersecting pieces, they can be slid into each other forming stable configurations without any gluing or additional connectors. These constructions enable quick prototyping and easy exploration of shapes, and are particularly useful for education. We propose a constraint-based optimization method and computational design framework to facilitate such structures. Planar surfaces can be connected, without any other materials or bindings, by cutting slits of the width of the material in the direction of the surface normal. These constructions are often found in cardboard and wooden 3-D puzzles (see Fig. 2). As laser cutters do not permit cuts that are not orthogonal to the surface, the slits would need to be larger to accommodate intersecting pieces at non-right angles (see Fig. 3). This would lead to unstable configurations and the need for connecting structures. Our goal is to avoid this complication and retain the simplicity of orthogonally intersecting pieces. As shown in Fig. 3, when considering a pair of intersecting orthogonal pieces, each piece has one degree of freedom in rotation. This introduces a constrained design space for composing more complex shapes from multiple interconnected pieces. The common method for generating such objects is using parallel or semantic cross-sections (as in Fig. 2), but this severely limits the design space. Slightly more complex methods grow the structure generatively, starting with a skeleton and ensuring orthogonality step by step, in either a manual or procedural fashion. However, these approaches quickly become infeasible due to combinatorial complexity