MiGumi: Making Tightly Coupled Integral Joints Millable

TL;DR

This work tackles the problem of fabricating traditional, tightly coupled integral joints with CNC milling, where nonzero tool radii erode sharp internal corners and disrupt precise surface contact. It introduces Millable Extrusion Geometry (MXG), a domain-specific language that represents parts as a stock minus a sequence of millable extrusion volumes, making artifact sources explicit and controllable. To recover tight coupling under fabrication constraints, the authors define two differentiable losses—Surface Gap and Milling Path Distance—and develop a co-optimization pipeline that operates on MXG representations, reducing the problem to planar slices for efficiency. They validate the approach on a dataset of 30 traditional joints (modeled in MXG0) and physically fabricate eight joints, showing CNC-compatible designs that preserve tight surface contact; the method also enables structured design exploration within milling constraints. Overall, MiGumi demonstrates that coupling-aware geometry can be reinterpreted for CNC workflows, expanding access to heritage joint designs while maintaining fabrication feasibility and functional fit.

Abstract

Traditional integral wood joints, despite their strength, durability, and elegance, remain rare in modern workflows due to the cost and difficulty of manual fabrication. CNC milling offers a scalable alternative, but directly milling traditional joints often fails to produce functional results because milling induces geometric deviations, such as rounded inner corners, that alter the target geometries of the parts. Since joints rely on tightly fitting surfaces, such deviations introduce gaps or overlaps that undermine fit or block assembly. We propose to overcome this problem by (1) designing a language that represent millable geometry, and (2) co-optimizing part geometries to restore coupling. We introduce Millable Extrusion Geometry (MXG), a language for representing geometry as the outcome of milling operations performed with flat-end drill bits. MXG represents each operation as a subtractive extrusion volume defined by a tool direction and drill radius. This parameterization enables the modeling of artifact-free geometry under an idealized zero-radius drill bit, matching traditional joint designs. Increasing the radius then reveals milling-induced deviations, which compromise the integrity of the joint. To restore coupling, we formalize tight coupling in terms of both surface proximity and proximity constraints on the mill-bit paths associated with mating surfaces. We then derive two tractable, differentiable losses that enable efficient optimization of joint geometry. We evaluate our method on 30 traditional joint designs, demonstrating that it produces CNC-compatible, tightly fitting joints that approximates the original geometry. By reinterpreting traditional joints for CNC workflows, we continue the evolution of this heritage craft and help ensure its relevance in future making practices.

Figures (16)

• Figure 1: The path towards a millable & usable dovetail joint. (a) The original joint geometry contains features that cannot be milled as they are incompatible with the CNC machine's finite tool radius. (b) Individually adapting each part to be millable introduces overlaps that prevent successful assembly. (c) Our method jointly optimizes all contact surfaces to produce geometry that is both millable and tightly coupled.
• Figure 2: Overview: Our method addresses two challenges: modeling milling-induced distortions and restoring tight coupling. Section \ref{['sec:millable_geom']} introduces MXG, a representation that models geometry as the outcome of milling operations, exposing rounding artifacts caused by milling. Section \ref{['sec:tight_coupling']} presents an optimization scheme that adapts joint geometry to restore coupling despite these artifacts.
• Figure 3: Morphological opening is the canonical way to derive geometry that can be safely removed by a milling tool. Given an implicitly defined input shape $f_i$ (left) and a structuring element $B_r$ representing the tool shape, the operation first applies a Minkowski subtraction (erosion, middle column) and then restores the removed volume by a subsequent addition (dilation, right column). The resulting shape excludes regions that the tool cannot reach and serves as a conservative, millable approximation of the original geometry.
• Figure 4: Extrusion with $r=0$
• Figure 5: We construct a millable extrusion by (a) defining a 2D SDF $f_i$ using a symbolic CSG tree, (b) applying morphological opening to obtain a millable base region $C_i^r$, and (c) extruding this region along direction $\mathbf{n}_i$ to produce the solid volume $\overline{\mathcal{E}_i}$.