Abstract
<title>Abstract</title> <p>Interpolatory subdivision generates smooth curves from control polygons by repeatedly inserting new points, and the classical six-point Deslauriers--Dubuc scheme is prized for producing curves that are not merely smooth but fair. We explain why through an exact local identity: on a six-point configuration whose neighbouring insertions are fixed by quintic interpolation, the Deslauriers--Dubuc midpoint is the unique minimiser of a discrete energy penalising the variation of second differences, which is the linearisation of the classical curvature-variation measure of fairness. The identity extends, within Kobbelt's variational subdivision framework, to the entire family. We then study the scheme's established Riemannian analogue, Wallner's geodesic-midpoint lift, which preserves the full Euclidean smoothness of the limit curves. We give explicit, self-contained implementations on the sphere and the hyperbolic plane, an elementary bound relating the lifted and planar rules, and a curvature-variation model on curved surfaces with closed-form solutions. On a benchmark suite, the six-point scheme balances fairness, locality, and robustness better than its four-point and eight-point siblings.</p>