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Charles Loop developed triangular subdivision surfaces. Loop subdivision repeatedly refines a control mesh and updates vertices by weighted averages, producing a smooth limiting shape.^[ref]

His later work modifies edge masks to obtain bounded curvature and a convex hull preservation. The analysis depends on eigenvalues of the subdivision operator, especially near extraordinary vertices whose number of neighbors differs from the regular triangular-grid case. Thus a local averaging algorithm has global smoothness and shape-preservation questions.^[ref]

Loop subdivision refines an interior triangular mesh by splitting each face into four triangles and updating control points with local affine combinations. For an old vertex $P$ with $n$ cyclic neighbors $P_1,\dots ,P_n$, the original vertex rule isFor an interior edge with endpoints $A,B$ and opposite triangle vertices $C,D$, the new edge vertex isOn the regular triangular lattice $n=6$, ${\beta }_6=1/16$, so $P^{\prime }=(10P+{\sum }_j{P}_j)/16$. The coefficient sums are one, giving affine equivariance of a geometric basis; their nonnegativity also gives a convex hull bound. These are interior rules: a boundary or a deliberately retained sharp crease needs its own specified stencil.

The scheme is approximating rather than interpolatory. If an old vertex has height one and its six regular neighbors have height zero, its new height is $10/16$, so refinement moves the old vertex. On regular patches the limit is a quartic box spline; extraordinary subdivision vertices require the local subdivision matrix and characteristic map of a subdivision surface analysis. The method is named for Charles Loop, whose 1987 thesis introduced the triangular scheme.^[ref][ref]

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