64 lines
2.6 KiB
Markdown
64 lines
2.6 KiB
Markdown
# SDF Tricks Detailed Reference
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## Prerequisites
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- Understanding of signed distance fields and ray marching
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- Basic SDF primitives and boolean operations
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- FBM / procedural noise fundamentals
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## Lipschitz Condition and FBM Detail
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An SDF must satisfy the **Lipschitz condition**: `|f(a) - f(b)| ≤ |a - b|` (gradient magnitude ≤ 1). This guarantees that stepping by the SDF value is always safe — no surface exists within that radius.
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When adding FBM noise to an SDF, the noise derivatives can violate Lipschitz:
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- Raw noise amplitude of 0.1 with frequency 20 has gradient ~2.0, breaking the condition
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- This causes ray marching to overshoot, creating holes and artifacts
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**Solutions**:
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1. **Amplitude limiting**: Keep `amplitude × frequency < 1.0` across all octaves
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2. **Distance fade**: `d += amp * fbm(p * freq) * smoothstep(fadeStart, 0.0, d)` — detail only appears near the surface where overshoot distance is small
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3. **Step size reduction**: Multiply ray step by 0.5-0.7, trading speed for stability
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## Bounding Volume Strategies
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### Hierarchical Bounding
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For scenes with N objects, test bounding volumes in order of increasing cost:
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```
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Level 1: Scene bounding sphere (1 evaluation)
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Level 2: Object group bounds (few evaluations)
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Level 3: Individual object SDF (full cost)
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```
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### Spatial Partitioning
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For repeating structures, combine domain repetition with bounds:
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```glsl
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float map(vec3 p) {
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vec3 q = mod(p + 2.0, 4.0) - 2.0; // repeat every 4 units
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// Only evaluate detail if within local bounding sphere
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float bound = length(q) - 1.5;
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if (bound > 0.2) return bound;
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return detailedSDF(q);
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}
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```
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## Binary Search Convergence
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After N iterations of binary search, the position error is `initialStep / 2^N`:
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- 4 iterations: 1/16 of initial step size
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- 6 iterations: 1/64 of initial step size (sub-pixel at typical resolutions)
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- 8 iterations: 1/256 (overkill for most uses)
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6 iterations is the practical sweet spot — gives sub-pixel precision without wasting GPU cycles.
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## XOR Operation Mathematics
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`opXor(a, b) = max(min(a, b), -max(a, b))`
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This is equivalent to: `union(a, b) AND NOT intersection(a, b)` — the symmetric difference. Geometry exists where exactly one shape is present but not both. Useful for creating lattice structures and interlocking patterns.
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## Interior SDF Pattern Techniques
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When the camera is inside an SDF (d < 0), the negative distance still gives useful information:
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- `abs(d)` gives distance to nearest surface from inside
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- Combine with repeating patterns using `fract()` to create infinite interior structures
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- Use `max(outerSDF, innerSDF)` to confine interior patterns within the outer shell
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