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 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.
Traditional integral joints are masterpieces of woodworking that have stood the test of time. Yet creating them by hand demands years of practice, specialized tools, and expert craftsmanship—making them infeasible for most carpenters to adopt.
Computational fabrication offers a promising alternative: users design the joint they envision, and let the computer handle fabrication. For wood, CNC milling is the method of choice—it's fast, accessible, and preserves the material's structural integrity unlike 3D printing.
However, a fundamental challenge arises: traditional joints have sharp internal corners that cylindrical mill bits cannot produce. Naively milling these designs introduces rounding errors that break tight coupling, creating gaps or overlaps that prevent assembly. This is illustrated in the accompanying figure.
We overcome this by (1) designing a language that represents millable geometry, and (2) co-optimizing part geometries to restore coupling.
We introduce Millable eXtrusion Geometry (MXG), a representation that models geometry as the outcome of milling operations performed with flat-end, uniform-radius tools. MXG is millable by design and parameterized by the tool radius, making it natural for analysis and mitigation of tool-induced artifacts.
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 (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.
Building on morphological opening, we construct millable extrusions through a three-step process. First, we (a) define a 2D signed distance function (SDF) using a symbolic constructive solid geometry (CSG) tree, which provides rich parametric control over the profile shape. Next, we (b) apply morphological opening to this profile to obtain a millable base region that respects the tool radius constraints. Finally, we (c) extrude this region along the milling direction to produce the solid subtractive volume. This construction pipeline ensures that every extrusion is millable by design.
With millable extrusions as primitives, we can now model complete traditional integral joints in MXG. Each part is represented as a material stock minus a sequence of subtractive extrusions. The figure shows several examples of traditional joints, each visualized with its resulting solid geometry alongside the milling directions (arrows) and 2D contours of the subtractive extrusions. These examples are designed with an idealized tool radius of r=0, capturing the original sharp-cornered geometry of traditional designs before accounting for real-world milling constraints.
A key feature of MXG is its ability to explicitly model and control milling-induced artifacts through two critical parameters: the drill radius r and the subtraction direction. The left column shows a target part geometry designed with an idealized tool radius (r=0). As we vary these parameters, the artifacts manifest in predictable ways: (a) increasing the drill radius r progressively amplifies corner rounding, and (b) varying the milling direction shifts both the location and orientation of these artifacts. This explicit parameterization allows us to analyze how real-world milling constraints will affect the joint geometry.
Once we can model milling artifacts, the challenge becomes restoring tight coupling. Increasing the drill radius in an MXG program from r=0 to a realistic value introduces rounding that creates gaps or overlaps. We address this by co-optimizing all parts simultaneously to restore precise contact while remaining millable.
(a) Our optimization pipeline exploits a structural property of MXG representation to greatly reduce the problem complexity. Based on its definition as a composition of subtractive planar extrusions, we observe that tight coupling can be fully characterized in 1D slices that lie perpendicular to the extrusion directions. (b) Many of the slices exhibit identical behavior and can be further grouped into a few representative planar sets. (c) The optimization domain then reduces to 1D planar curves within these representative slice sets. The interface of Z1’s slice with extrusions from other directions is shown as a dashed red line. Such boundaries are kept fixed. (d) We optimize this reduced space to compute tightly coupled geometry, and (e) reassemble the results into the final 3D MiGumi parts.
We use two complementary measures for tight coupling. Surface Gap (middle) integrates closest-point distances along planar curves to ensure mating surfaces remain in contact. However, optimizing surfaces alone is unstable since different mill paths can produce similar surfaces. Milling Path Distance (right) addresses this by constraining tool path spacing itself, penalizing deviations from the ideal distance of 2·r. Together, these enable stable gradient-based optimization maintaining both surface contact and fabrication validity.
MXG enables structured exploration of design alternatives. By explicitly modeling the parameters that induce milling artifacts—tool direction and drill radius—authors can evaluate how different configurations trade off aesthetic, structural, and fabrication considerations.
For certain joints, multi-directional milling is critical to ensure that tight coupling is feasible. First, the use of general CSG-like 2D expressions enable us to cover a wider spread of designs. Second, for certain joints, multi-directional milling is critical to ensure that tight coupling is feasible.
Our dataset consists of 30 traditional integral joint designs, each modeled parametrically using our MXG representation. These parametric programs allow for continuous variation in dimensions, proportions, and milling configurations—enabling the synthesis of a much larger family of design variants from each base joint. We plan to expand the dataset to include additional designs from historical catalogs and contemporary applications.
@article{ganeshan2025migumi,
author = {Ganeshan, Aditya and Fleischer, Kurt and Jakob, Wenzel and Shamir, Ariel and Ritchie, Daniel and Igarashi, Takeo and Larsson, Maria},
title = {MiGumi: Making Tightly Coupled Integral Joints Millable},
year = {2025},
issue_date = {December 2025},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {44},
number = {6},
issn = {0730-0301},
url = {https://doi.org/10.1145/3763304},
doi = {10.1145/3763304},
journal = {ACM Trans. Graph.},
month = {dec},
articleno = {193},
numpages = {17}
}We would like to thank the anonymous reviewers for their helpful suggestions. Aditya Ganeshan conducted part of this work as a visiting researcher at the Igarashi Laboratory, University of Tokyo.