Wire cut of double-sided minimal surfaces

Abstract

We present a systematic method for producing double-sided minimal surfaces by wire-cut machines. A link between minimal surfaces and ruled surfaces is pursued through wire cutting. Weierstrass parameterization is employed to define minimal surfaces ( $$\mathbb {R}^3$$ ) over a complex plane ( $$\mathbb {C}$$ ). Our method consists of three components. First, the orthogonal double-sided cuts match a pair of orthonormal tangent vectors on the surface. Second, A closed-form expression for the principal directions facilitates the global quadrangulation of minimal surfaces. Third, the CNC machine’s toolpath results from the surface’s analytic characterization. Asymptotic cutting and principal cutting are compared in terms of collisions and cutting error. We employed a general-purpose language (Java) to create machine instructions from the Weierstrass representation of minimal surfaces. Thus, the entire workflow from mathematical modeling to production involves no 3D models or CAD/CAM software. Both a 5-axis wire cutter and a customized robotic system were tested.