Slide each projection point along its circle to minimize L·k/L₀ + w·T/(2πk) (or the T term alone with Turning only): L per mean loop perimeter and T per full turn.

With Use rope, L and T are the rope's instead of the contour's: the projection points joined in contour order, straight or through the corner of their tangents (see Rope flat angle), else along the contour without cutting into a circle; its T counts every vertex.

L: contour length; with Length set to non-tangent edges, without the edges between a projection point and its own circle (squared: Σ l²).
T = Σ_i c_i·|θ_i|^p (p = Turning power; p < 1 favors few sharp corners), θ_i the turning angle at contour vertex i (skipped on a projection point, except on the rope).
c = min(1, (l_a+l_b)/(2h)), l_a, l_b the lengths of its two edges.

Weight: w. Right of it: T of the shown contour (or rope), and the plain Σ|θ| without h.
Turning scale h: × sample spacing.

Loss (the refine's objective, for the current points): L·k/L₀ + w·T/(2πk), L₀ and k the length and number of loops of the contour (or rope) where the refine started.