Human or algorithmic designers of laser-formable structures must make the decision on where (which edges of) the 3D goal shape, represented as a surface mesh, should be cut so the flattened 2D net is laser-formable. Consequently, our proposed computational framework adopts an iterative process of design and evaluation. Because a 3D surface mesh can be flattened onto many nets and a net can be folded in many ways, the evaluation step must plan and simulate the entire fabrication process. Compared with a self-folding net  (?), lasergami encounters different challenges from its aforementioned special constraints. Combined with the inherent high dimensionality of the design space that inhibits analytical solutions, these differences make manual design nearly impossible.

Two algorithms devised to overcome these challenges of lasergami are nearly-blooming (NB) net generation and as-folded-as-possible (FP) motion planning. We introduce these through the manufacturing flowchart in Fig. 2 and in the following sections, providing additional details in the supplementary materials. In Fig. 2, the NB algorithm cuts the goal mesh along its edges followed by motion planning that finds the folding motion satisfying the remaining constraints. If such a folding motion path does not exist, the planner finds a path to bring the structure as close to the fully folded configuration as possible. Additional optimization (Fig. 2(D)) then finds slightly altered nets that still satisfy all the lasergami constraints with overall improved performance. The substrate, which has its own mesh (Fig. 2(E)), has $F$ faces clipped or folded (Figs. 2(F), S14, S15) to ensure the net does not collide with the substrate (Alg. S2). Finally, instructions for the laser beam (path, speed, power) are generated and passed to the laser cutter (Fig. 2(G)). Each step handles different but interdependent aspects of the manufacturing process; thus, the influence of early steps on the performance of the next ones must be carefully assessed. For example, we deliberately ensure that the NB unfolding can reduce chances of self-collisions during folding (Fig. S13) and areas of clipped substrate (Fig. S14). Figure 2(B) shows an example of a NB unfolding which, when compared with a net created from genetic algorithms (GA), can be laser folded closer to the goal (97.2% vs. 68.6%).

Blooming unfolding imitates flower blooming motion which continuously unfolds a polyhedron onto a foldable net without collisions  (?). Consider a function $u_{B}:G\rightarrow N$ that maps a goal shape $G$ to a blooming net $N$. The collision-free motion of $u_{B}$ implies that the volume swept by unfolding $G$ to $N$ is smaller than other nets. Thus, a blooming net reduces the possibility of collision with the laser line of sight and substrate, making it the best option for lasergami. Although it is known that all convex shapes have blooming unfoldings  (?), applying $u_{B}$ to non-convex shapes, in general, does not create a net that admits continuous blooming  (?). In fact, as illustrated in Fig. 2(B), when $G$ is not convex, $u_{B}(G)$ is most likely an unfolding containing overlaps, i.e., not a laser-formable net. Therefore, we propose the idea of a NB net (Alg. S1) that unfolds $G$ into a net with the provably fewest “edits” from a blooming unfolding (Thm. S1-S2).

A motion planner instructs step-by-step where and when the laser should be emitted so the resulting folding motion satisfies all constraints. Solving a motion planning problem requires significant computation, but there are several reasons that make planning necessary: (A) The nearly-blooming net lowers the chance of collisions during folding but does not eliminate them. (B) While some laser occlusions can only be resolved by modifying the nets (Fig. S6), many laser occlusions are avoidable by folding some creases before others under the outside-in folding order (Fig. 1(B)). Because of the outside-in folding order, one might suspect that the configuration space of a net is highly constrained and low-dimensional. However, we proved that the planning still requires time complexity exponential to the number of faces in the net (Thm. S3).

When a net cannot be fully folded, we observe that many collisions and occlusions can be avoided if we can tolerate that some creases are not fully folded, leading to a second key technique for enabling lasergami: a new folding motion planner that can fold a net as close to its goal as possible by unfolding folded creases as little as possible. Fig. 2(C) shows, in comparison to the traditional motion planner (MP)  (?), FP motion planning can fold the net significantly closer to the goal shape (26.25% vs. 97.20%).

The main advantages of the proposed NB unfolding and FP planner were explored for a diverse set of goal shapes, some of them portrayed in Fig. 3. All NB nets folded by the FP planner show superior laser formability averaging 96.08% fold completion, compared with GA nets folded by the FP planner, NB nets folded by the traditional MP, and GA nets folded by the traditional MP, having 75.47%, 46.87%, and 13.31% average fold completions respectively (Fig. S13). The fold completion results for three goal shapes (bunny, whale, and snail) are shown in Fig. 3. Additionally, we observe that NB nets clip only 71.80% of substrate area compared to nets created using GA (Fig. S14).

The proposed process is highly flexible, and additional optimization may further improve performance  (?) by reducing energy, error, or the number of faces that require clipping as shown in Fig. 2(D). Significant $1/10$th fabrication energy $E$ was saved for the bunny shape using additional optimization, with only $F=11$ faces requiring clipping compared to the initial $49$ faces. The FP motion planner may also integrate approximations of the thermal field to enable even faster laser manufacturing  (?).

Without the assistance from computational design, human’s cognitive capability is limited to creating simple lasergami structures. This work unleashed the potential of lasergami and demonstrated a diverse set of metallic shapes previously believed to have been unfeasible to be laser-formed. This contribution was enabled by new computational methods, namely nearly-blooming unfolding and as-folded-as-possible planning. While the novel proposed methods showed effectiveness and allowed for rapid, straightforward, and cost-effective fabrication of 3D shapes using a budget laser cutter, the structural complexity, fabrication precision, and time efficiency can be further increased using more advanced laser cutters with more degrees of freedom.