Gradients are the new intervals

A deep dive into using Lipschitz continuity and gradient bounds as an alternative to interval arithmetic for implicit surface rendering. The author explores how single-point evaluation on Lipschitz-continuous distance fields can produce 'pseudo-intervals' that replicate the pruning and simplification benefits of interval arithmetic, often with fewer issues around stacked transforms. The post covers basic rendering, the advantages and downsides of each approach, normalization strategies for non-Lipschitz fields (using forward-mode automatic differentiation with per-branch normalization before min/max nodes), and expression simplification. Benchmark comparisons show interval arithmetic is slightly faster than normalized pseudo-intervals for a large test case, but the gap is modest.