definition. sphere tracing [hart1996sphere, sec. 2.3, eq. 12] [ag-001I]

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The sphere tracing algorithm is to march a ray by an adaptive safe step size, which is the absolute value of the signed distance bound calculated by \(F/\lambda \) per theorem [ag-000T], the rest is the same as ray marching (naïve).

Formally, the root found by sphere tracing is the limit point of the sequence defined by the recurrence equation \[t_{i+1} = t_i + \frac {|F(t_i)|}{\lambda } = t_i + \frac {f(\boldsymbol {r}(t_i))}{\lambda } \] where \(F\) is the ray intersection, and \(f\) is the SDF.

Usually \(\lambda = 1\), and \(F(t_i)\) stays positive until the stopping condition \(F(t_i) < \epsilon \) is met, so the above can be simplified to \[t_{i+1} = t_i + F(t_i) = t_i + f(\boldsymbol {r}(t_i)) \]