trueform

About | TS

Research

The theory and publications behind trueform's algorithms.

Geometry pipelines accumulate defects. Sculpting, remeshing, format conversion—each step can introduce non-manifold flaps, inconsistencies, and surface artifacts. By the time geometry reaches your algorithm, it is rarely the ideal manifold most methods assume.

trueform is built around a simple idea: a mesh should be understood as an intended surface, observed through the noise of finite precision and prior processing.

These defects are additive—they attach to the surface without creating holes. The underlying structure remains recoverable.

\mathcal{M}':\left\{\;\begin{smallmatrix} T' \\ G' \end{smallmatrix}\;\right\}

\mathcal{M}:\left\{\;\begin{smallmatrix} T' \cup T_\Delta \\ G' + G_\Delta \end{smallmatrix}\;\right\}

\cup\, \textcolor{#14b8a6}{T_\Delta} \sim \mathcal{D}_T

+\, \textcolor{#14b8a6}{G_\Delta} \sim \mathcal{D}_G

\mathcal{F}

Processing adds artifacts to the underlying mesh $\mathcal{M}'$ : topological noise $\textcolor{#14b8a6}{T_\Delta}$ (flaps, bad winding) and geometric noise $\textcolor{#14b8a6}{G_\Delta}$ (displaced vertices). Idealization $\mathcal{F}$ recovers $\mathcal{M}'$ from $\mathcal{M}$ .

This perspective shapes the algorithms:

Geometric exactness. Input coordinates are scaled to exact integer space (int32 or int64). All predicates—orientation tests, intersection points, determinants—are computed with exact integer arithmetic through the meta<Int> type chain, eliminating floating-point error from geometric decisions.
Topological exactness. The full 5-type intersection classification (VV, VE, EE, VF, EF) resolves every degenerate configuration into its canonical form. Where multiple intersection curves cross on a face, implicit predicates detect and resolve the crossings—splitting curves at crossing points so the face subdivision sees clean topology.
Artifacts are modeled as independent noise. The quantization from float to integer coordinates can collapse near-coincident points or create degenerate slivers. These artifacts decompose locally across polygon regions; topological artifacts decompose globally across manifold edge-connected components—enabling robust classification of the intended structure through exact wedge predicates and majority voting.

The goal is commutative correctness: operations that commute with mesh idealization. Chain operations freely on non-ideal meshes and clean up once at the end.

Boolean operation $\mathcal{B}$ commutes with mesh idealization $\mathcal{F}$ . Top: $\mathcal{B}$ on a mesh with artifacts produces a result with artifacts; the intersection graph $\mathcal{G}_\mathcal{I}$ contains a branching edge. Bottom: $\mathcal{B}$ on idealized meshes produces a clean result; $\mathcal{G}_\mathcal{I}$ is a simple closed curve. Apply $\mathcal{F}$ at any point—both paths yield equivalent geometry.

Publications

tf::tree: A General-Purpose Spatial Hierarchy for Real-Time Geometry Queries. Žiga Sajovic, Dejan Knez, and Robert Korez. (IEEE), June 2025. https://doi.org/10.36227/techrxiv.174952959.92977743/v1
Real-Time Mesh Booleans that Commute with Mesh Idealization. Žiga Sajovic and Dejan Knez. (IEEE), May 2025. https://doi.org/10.36227/techrxiv.174667714.42575478/v1

Citation

If you use trueform in your work, please cite:

@software{trueform2025,
    title={trueform: Real-time Geometric Processing},
    author={Sajovic, {\v{Z}}iga and {et al.}},
    organization={XLAB d.o.o.},
    year={2025},
    url={https://github.com/polydera/trueform}
}

Raycast Rendering

Batch ray casting with Lambertian shading, producing a BMP image entirely with NDArray operations.

Contributing

Guidelines for contributing code, documentation, and examples.