= Solution
For a <polymer> curve parameterized by contour distance $s$, let $\mathbf t(s)$ be its unit tangent. \b[The <persistence length> is the characteristic contour distance over which tangent orientations lose correlation.] In the usual three-dimensional <worm-like chain> convention,
$$
\boxed{\langle\mathbf t(s)\cdot\mathbf t(0)\rangle=e^{-|s|/L_p},
\qquad L_p=\frac{\kappa_b}{k_BT},}
$$
where $\kappa_b$ is the bending stiffness. Segments much shorter than $L_p$ look approximately straight; a much longer chain can contain many statistically distinct orientations.
A surface image can be traced to estimate tangent correlations as a function of contour separation, after accounting for tracing resolution and substrate effects. A strictly equilibrated two-dimensional chain has a different relation between bending stiffness and correlation decay length from the three-dimensional convention. The definition must therefore state whether the surface preserves a projected conformation or allows in-plane equilibration. This is a statement about statistical filament mechanics; biological interpretation is deferred.
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