Subtractive Trajectory Generative Modeling: A Material-Removal-Centric Framework for Three-Dimensional Shape Generation
Abstract
Three-dimensional shape generation has been studied through a variety of representations and learning paradigms, including voxel-based generation, point-cloud synthesis, implicit neural representations, neural radiance fields, Gaussian-based scene representations, and procedural computer-aided design (CAD) generation. Despite their differences, many approaches primarily formulate generation as the direct synthesis or reconstruction of a target representation. This paper proposes Subtractive Trajectory Generative Modeling (STGM), a material-removal-centric framework that formulates three-dimensional shape generation as the progressive transformation of an initial stock into a target solid through a sequence of removal operations. Instead of generating the final geometry exclusively as a static representation, STGM models the evolving residual material state, the geometry of each removal region, the ordering of operations, and the compatibility of each operation with manufacturing constraints. The framework combines a geometric state representation, a conditional removal policy, differentiable or approximately differentiable geometric updates, and a multi-objective optimization procedure. Its objectives include target-shape fidelity, preservation of required material, operation complexity, accessibility, and estimated manufacturing cost. A hierarchical strategy is proposed to organize coarse removal, intermediate structural formation, and fine geometric refinement. The paper defines a formal problem, presents a reference algorithm, specifies a training objective, and develops an experimental protocol for evaluating geometric accuracy, trajectory quality, manufacturability, and computational efficiency. STGM is presented as a research hypothesis rather than an empirically validated method. Its potential contribution lies in integrating material-state evolution and removal-sequence optimization into a unified generative formulation, while its novelty and effectiveness require systematic comparison with existing CAD sequence generation, constructive solid geometry, machining-aware modeling, and shape-generation approaches.
Related
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BibTeX
@misc{zhang2026subtractive,
title = {{Subtractive Trajectory Generative Modeling: A Material-Removal-Centric Framework for Three-Dimensional Shape Generation}},
author = {Jincheng Zhang},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23636236},
url = {https://zenodo.org/records/23636236},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Subtractive Trajectory Generative Modeling: A Material-Removal-Centric Framework for Three-Dimensional Shape Generation
AU - Jincheng Zhang
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23636236
UR - https://zenodo.org/records/23636236
AB - Three-dimensional shape generation has been studied through a variety of representations and learning paradigms, including voxel-based generation, point-cloud synthesis, implicit neural representations, neural radiance fields, Gaussian-based scene representations, and procedural computer-aided design (CAD) generation. Despite their differences, many approaches primarily formulate generation as the direct synthesis or reconstruction of a target representation. This paper proposes Subtractive Trajectory Generative Modeling (STGM), a material-removal-centric framework that formulates three-dimensional shape generation as the progressive transformation of an initial stock into a target solid through a sequence of removal operations. Instead of generating the final geometry exclusively as a static representation, STGM models the evolving residual material state, the geometry of each removal region, the ordering of operations, and the compatibility of each operation with manufacturing constraints. The framework combines a geometric state representation, a conditional removal policy, differentiable or approximately differentiable geometric updates, and a multi-objective optimization procedure. Its objectives include target-shape fidelity, preservation of required material, operation complexity, accessibility, and estimated manufacturing cost. A hierarchical strategy is proposed to organize coarse removal, intermediate structural formation, and fine geometric refinement. The paper defines a formal problem, presents a reference algorithm, specifies a training objective, and develops an experimental protocol for evaluating geometric accuracy, trajectory quality, manufacturability, and computational efficiency. STGM is presented as a research hypothesis rather than an empirically validated method. Its potential contribution lies in integrating material-state evolution and removal-sequence optimization into a unified generative formulation, while its novelty and effectiveness require systematic comparison with existing CAD sequence generation, constructive solid geometry, machining-aware modeling, and shape-generation approaches.
ER -