Lipschitz bounds

Hart’s paper is the clearest foundational account of why sphere tracing works at all. It does not present marching as a heuristic loop, but as a geometric method built on distance bounds: if the field gives a trustworthy lower bound on distance to the surface, each step is guaranteed not to cross the zero set. The paper also frames this against the weaknesses of blind sampling and derivative-heavy root-finding methods, which is useful if you want to explain why sphere tracing became the robust default for many implicit scenes.

It is also stronger than most practical tutorials on the mathematical side. Hart ties the method to Lipschitz conditions, explains why a bound on derivative magnitude is enough even when derivatives are discontinuous or undefined, and connects the technique to antialiasing through cone-tracing-style reasoning. On top of that, the appendices broaden the paper from a rendering method into a modeling reference by deriving distance functions and operations for many important implicit primitives and deformations.