Hatching for 3D Prints: Line-Based Halftoning for Dual Extrusion FDM

1. Introduction

This paper addresses a significant gap in Fused Deposition Modeling (FDM) 3D printing: the ability to produce objects with the appearance of continuous grayscale or color imagery. While inkjet-based additive manufacturing systems offer color, FDM techniques have been limited, often sacrificing surface quality, geometric integrity, or introducing long print times. The work presents a novel line-based halftoning technique, termed "hatching," specifically designed for dual-extrusion FDM printers. This method modulates the visible width of printed lines from two different colored materials to create the perception of grayscale gradients, without adversely affecting the core printing process or the structural properties of the final object.

2. Methodology

The proposed technique adapts the 2D printing concept of hatching—using lines of varying spacing or thickness to simulate tone—to the 3D layer-by-layer context of FDM.

2.1. The Hatching Principle

Instead of using discrete dots (as in traditional halftoning), this method uses the continuous extrusion paths inherent to FDM. By alternating between two materials (e.g., black and white) within a single layer and controlling their relative widths, the perceived local grayscale tone is achieved. A key innovation is orienting these hatched lines to be locally perpendicular to the viewer's likely sightline, optimizing the effect for curved and sloped surfaces.

2.2. Implementation for FDM

The algorithm is integrated into the slicing process. For each layer, the surface geometry is analyzed. Grayscale image data is mapped onto the surface. The toolpath is then generated to interweave filaments from two nozzles, with the extrusion width for each color modulated according to the target grayscale value at that location. The implementation is open-source within the Ultimaker CuraEngine.

3. Technical Details & Mathematical Model

The core of the technique is a mapping from a desired grayscale intensity $I$ (where $0 \leq I \leq 1$, with 0 being black and 1 being white) to the physical widths of the two extruded lines. For a given hatch line, if $w_{total}$ is the total width allocated for one cycle of the two materials, the width of the "foreground" material (e.g., black) $w_f$ and the "background" material (e.g., white) $w_b$ can be defined as:

$w_f = I \cdot w_{total}$

$w_b = (1 - I) \cdot w_{total}$

The perceived tone $T$ is a function of these widths and the viewing angle $\theta$, approximating the projected visible area of each color: $T \approx f(w_f, w_b, \theta)$. The algorithm aims to solve for the toolpath that achieves a target $T$ across the surface.

4. Experimental Results & Analysis

Experiments were conducted on a dual-nozzle FDM printer using black and white PLA filaments.

4.1. Test Prints & Visual Assessment

The paper showcases several demonstrative prints (referenced in Figure 1 of the PDF): a 3D portrait, an artistic figurine, a soda can with text, and a connecting rod with stress analysis visualization. The results show a clear perception of grayscale gradients on both vertical and moderately sloped surfaces. High-frequency details from the source images are preserved more effectively than in previous low-frequency texture modulation techniques.

4.2. Performance Metrics

Print Time Impact

Minimal increase compared to a solid single-color print, as the technique primarily modifies toolpaths within a layer rather than adding layers or complex movements.

Geometric Fidelity

Surface geometry is largely preserved, unlike methods that deposit extra material or create surface textures. The primary change is visual, not topological.

Limitation on Shallow Slopes

The halftoning effect degrades on surfaces approaching horizontal, as the line-based pattern becomes less visible from a top-down perspective.

5. Analysis Framework: Core Insight & Critique

Core Insight: Kuipers et al. have executed a brilliant lateral move. They stopped trying to force droplet-based halftoning onto a line-based manufacturing process (a square-peg-round-hole problem that plagues FDM color research) and instead embraced the line as the fundamental pixel. The core insight isn't a new algorithm, but a reframing: the extrusion path is the native display element. This aligns with the philosophy seen in advanced image synthesis, where the representation defines the possibility space (e.g., Neural Radiance Fields (NeRF) using continuous volumetric scenes instead of discrete pixels).

Logical Flow: The logic is admirably clean: 1) Identify FDM's constraint (continuous paths), 2) Find a matching halftoning paradigm (hatching), 3) Map grayscale to line-width modulation, 4) Orient lines for optimal viewing. It bypasses the computational nightmare of simulating droplets, focusing on a control parameter (extrusion multiplier) already present in the slicer.

Strengths & Flaws: The strength is its elegant practicality—minimal process disruption, open-source implementation. Its major flaw is its nascency: it's a monochrome (grayscale) solution in a world that thinks in RGB. The paper admits the lack of perceptual calibration; a 50% gray might not look 50% gray due to material gloss and light scattering. Furthermore, it inherits all the alignment and oozing challenges of dual extrusion, which can blur the crisp line edges essential for the effect.

Actionable Insights: For researchers, the immediate next step is perceptual calibration using a methodology similar to color management in 2D printing (ICC profiles). For industry, this technique is ready for integration into slicers for functional grayscale printing (e.g., stress maps, depth codes). The real strategic play is to view this not as an end, but as a foundational layer. The logical extension is a CMYK hatching system, using the same line-width modulation principle for each color channel. The challenge won't be the algorithm, but the material science: developing filaments with reliable opacity and colorfastness for thin, overlapping extrusions.

6. Future Applications & Research Directions

Full Color Expansion: The most direct path is extending the model to three or four colors (CMYK). This would involve solving for overlapping hatched lines of different colors, a significant computational and material challenge.
Perceptual Calibration & Texture: Future work must establish a robust colorimetric model for filament pairs under various lighting conditions. Research could also explore modulating line height or texture in conjunction with width for enhanced tonal range.
Beyond Aesthetics - Functional Gradients: The principle can be applied to create objects with graded material properties. For example, modulating the ratio of a flexible filament to a rigid one along a toolpath could create parts with spatially varying stiffness, useful in soft robotics or ergonomic grips.
Integration with Volumetric Data: Direct printing of medical scan data (CT, MRI) as physical, tone-represented models for surgical planning, using grayscale to represent density or tissue type.

7. References

Kuipers, T., Elkhuizen, W., Verlinden, J., & Doubrovski, E. (2018). Hatching for 3D prints: line-based halftoning for dual extrusion fused deposition modeling. Computers & Graphics.
Ultimaker. (2018). CuraEngine. GitHub repository. https://github.com/Ultimaker/CuraEngine
Reiner, T., et al. (2014). [Reference to prior work on FDM color textures].
Mildenhall, B., et al. (2020). NeRF: Representing Scenes as Neural Radiance Fields for View Synthesis. ECCV. (Conceptual reference for representation defining possibility space).
International Color Consortium (ICC). (n.d.). Specification ICC.1:2022. https://www.color.org (Reference for color management systems).

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