Infill & Toolpaths

2 min read•1 page

Triply Periodic Minimal Surfaces (Gyroid Infill): Traditional rectilinear grid infills are anisotropic, failing under shear and transverse tension. The Gyroid is a non-self-intersecting minimal surface discovered by Alan Schoen (1970). Sliced along the Z axis, its undulating sinusoidal paths never cross on the same layer, delivering isotropic mechanical stiffness and continuous nozzle extrusion.

Gyroid Geometric Properties:

1. Zero Mean Curvature: H = 0 at all points, minimizing localized stress concentrations. 2. Periodic Symmetry: Crystal cubic space group I4_1 32 repeating indefinitely across 3D space. 3. Open Interconnected Porosity: Continuous fluid channels ideal for heat exchangers, bone scaffolds, and lightweight aerospace cores.

python

1import math
2
3def gyroid_level_set(x, y, z, scale=1.0, iso=0.0):
4    """
5    Evaluates Triply Periodic Minimal Surface (TPMS) Gyroid equation:
6    f(x, y, z) = sin(x)*cos(y) + sin(y)*cos(z) + sin(z)*cos(x) - iso = 0.
7    Provides isotropic strength-to-weight ratio in all 3 spatial directions.
8    """
9    sx, sy, sz = x * scale, y * scale, z * scale
10    return (
11        math.sin(sx) * math.cos(sy) +
12        math.sin(sy) * math.cos(sz) +
13        math.sin(sz) * math.cos(sx) - iso
14    )

Infill Slice Z

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Lattice Frequency

2.00

3 min read•1 page

Continuous Toolpath Planning (Fermat Spirals): Every time an extruder nozzle halts, retracts filament, and rapid-travels to another island, print time increases and molten plastic oozes, creating surface stringing and seam zits. Connected Fermat spirals fill closed boundaries in a single, unbroken extrusion curve winding outwards and interleaving back to the center.

Retraction-Free Infill:

1. Fermat Geometry: Radial expansion proportional to √θ maintains constant toolpath line spacing. 2. Eulerian Connectivity: 100% of the slice infill is deposited in a single uninterrupted motion. 3. Extruder Dynamics: Keeps nozzle pressure steady, minimizing under-extrusion at seam restarts.

python

1import numpy as np
2
3def generate_fermat_spiral(r_max=2.0, num_turns=6, num_samples=300):
4    """
5    Generates a continuous, non-retracting Fermat spiral toolpath (r = a * sqrt(theta)).
6    Transitions seamlessly from outward winding to inward return without crossing itself.
7    """
8    max_theta = (2.0 * np.pi) * num_turns
9    theta = np.linspace(0, max_theta, num_samples)
10    
11    # Fermat outward branch: r = c * sqrt(theta)
12    r_out = r_max * np.sqrt(theta / max_theta)
13    x_out = r_out * np.cos(theta)
14    y_out = r_out * np.sin(theta)
15    
16    # Inward interleaved return branch shifted by pi
17    x_in = r_out[::-1] * np.cos(theta[::-1] + np.pi)
18    y_in = r_out[::-1] * np.sin(theta[::-1] + np.pi)
19    
20    # Connected continuous toolpath without nozzle lift
21    full_x = np.concatenate([x_out, x_in])
22    full_y = np.concatenate([y_out, y_in])
23    return list(zip(full_x, full_y))

Spiral Revolutions

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