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The characteristic map of a subdivision surface is the planar limit map obtained from the two independent tangent eigenvectors of its local subdivision matrix. In the usual stationary setting, a simple constant eigenvalue one, a semisimple double subdominant eigenvalue $0<\lambda <1$, strictly smaller other eigenvalues, and a regular injective characteristic map establish the standard $C^1$ tangent-plane behavior for nondegenerate data. The eigenvalue inequalities alone do not exclude a folded or singular characteristic map. For an alternating scheme these tests must use a full period of refinement.

An extraordinary subdivision vertex has a valence different from that of the regular refinement lattice. Regular box spline calculations do not alone establish smoothness at this vertex. The finite local subdivision matrix, its eigenvalues, and the associated characteristic map of a subdivision surface determine convergence and tangent behavior.

This is mid-edge subdivision, rather than a vertex-retaining refinement. Use the usual closed two-manifold mesh setting: each edge has two incident facets and each vertex has a cyclic fan. For boundary meshes, a boundary refinement rule has to be specified separately. The relevant generic-position condition excludes collapsed local tangent directions and singular regular patches.

Topology and the manifold property. Regard the abstract mesh as a cell decomposition of a surface. Join the edge midpoints in each facet. The inner polygon is the new face associated with that facet; the corner regions around an old vertex assemble into the face associated with that vertex. At an old edge midpoint, the four adjacent corner connections form a cyclic link. Thus every new interior vertex has valence four, every new edge has two incident faces, and the refined cell complex is homeomorphic to the original surface. If the original counts are $V,E,F$, the new counts are $E,2E,V+F$, preserving $V-E+F$. The cyclic-link argument, rather than Euler characteristic alone, verifies the local manifold condition. Convergence and local regularity, justified below, transfer this abstract surface structure to the limit.

Convex hull. Each newly created point is an average of two previous points. By induction it is a convex combination of original vertices. The same is true of points of any triangulation of the refined facets, and taking limits keeps them in the closed convex hull. Hence the entire limit stays in that hull, not necessarily inside the original nonconvex enclosed solid.

Facet centroids. Follow the descendant inner face of an original $n$-sided facet, with cyclic vertices $p_i$. Its vertices satisfy $p_i^{(r+1)}=(p_i^{(r)}+p_{i+1}^{(r)})/2$. Summing proves mid-edge facet centroid invariance:For a cyclic Fourier mode $e^{2\pi iji/n}$ the multiplier is $(1+e^{2\pi ij/n})/2$, whose modulus is $|\cos (\pi j/n)|<1$ for $j\ne 0$. Every mean-zero mode therefore decays, so all descendant face vertices converge to $\overline{p}$. Consequently the original vertex centroid belongs to the limit surface. No planarity assumption is needed.

Quadratic triangular pieces. On a regular quadrilateral mesh, apply two midpoint steps. Each new point combines three old neighboring controls with weights $1/2,1/4,1/4$. For example, around a face with cyclic corners $p_{00},p_{10},p_{11},p_{01}$, the four new points areThis is the binary refinement mask obtained by expandingwith the translation chosen to place the refined grid at quarter offsets. The limit is therefore represented by shifts of the quadratic four-direction box spline with directions $e_1,e_2,e_1+e_2,e_1-e_2$. Projection of a four-cube onto the plane has piecewise total degree $4-2=2$, and its knot grid is triangular. Coordinatewise linear combinations preserve that degree, giving parametric quadratic triangle patches. Around a nonregular center there are infinitely many shrinking rings of such patches; the isolated center is their limit, not generally part of a finite polynomial patch representation.

Only finitely many initially possible smoothness exceptions. Every vertex has valence four after the first step. Only the descendants of original facets and the faces created around original vertices can remain nonquadrilateral. All subsequently created faces around four-valent vertices are quadrilaterals. Thus every point except the limiting centers of those persistent nonquadrilateral faces eventually lies in a regular patch neighborhood. The number of possible exceptional centers is at most $V+F$. For the regular box spline, any three distinct directions still span the plane, so three directions must be removed to make the remainder nonspanning. The box spline continuity criterion gives $C^{3-2}=C^1$ across its triangle boundaries. This proves $C^1$ continuity away from at most those finitely many centers.

Removing the apparent exceptions. Center a finite neighborhood of an $n$-sided exceptional face, and analyze a full two-step subdivision matrix so the mesh orientation is restored. Cyclic Fourier basis modes diagonalize its sector symmetry. The face mode has eigenvalueThe local ring blocks also contribute eigenvalues $1/4$ and zero. For $n>3$, the constant mode is the single eigenvalue one, and the two first-frequency tangent modes dominate all nonconstant remaining modes. Construct their characteristic map of a subdivision surface; its quadratic derivative pieces must have a nonzero consistently oriented Jacobian, and its fundamental sector must be injective. These checks, not the eigenvalue ordering alone, give a regular chart and a continuously limiting tangent plane for generic data.

The case $n=3$ needs a separate Jordan analysis: $\lambda =1/4$ is eightfold, so the usual semisimple double-eigenvalue test does not apply. Its first-frequency Jordan chains supply the leading tangent behavior, while the other contributions become relatively negligible. Regularity and injectivity of that generalized characteristic map complete the argument. These concrete spectral and characteristic-map checks are established for this scheme in Peters and Reif, Sections 3 and Appendix. Thus the generic limit is geometrically $C^1$ also at the exceptional centers; compatible local charts need not preserve the original uniform refinement parameterization.

The manifold statement concerns the abstract limit surface and its regular local parametrizations. It does not, from combinatorics and generic tangent data alone, prove global injectivity of the image in three-dimensional space. A particular model may require an additional global self-intersection test before claiming that its image is an embedded manifold.

The corresponding subdivision surface interrogation needs mesh connectivity and patch adjacency, boundary and crease flags, local coordinate charts with transition maps, and subdivision into smaller patches with their original-chart mappings. Its evaluation enquiry returns a limit point, tangent derivatives and a normal vector; second derivatives are supplied only where the limit is sufficiently smooth. Certified bounding volumes and patch approximation errors allow a hierarchy for intersection and proximity searches. Boundary-curve enquiries complete the interface for clipping.

On regular parts of a subdivision surface, these evaluations may use its equivalent tensor-product or box spline patch. Near an extraordinary subdivision vertex, the local subdivision matrix determines the limit position through its constant left eigenvector, and its tangent modes determine the normal vector under the scheme's characteristic map of a subdivision surface regularity conditions. Refinement alone does not establish those conditions. A subdivision surface may have a well-defined tangent plane there without possessing second derivatives or Gaussian curvature. An interrogation interface must distinguish these cases and can work on surrounding regular charts when a higher derivative is unavailable at the vertex itself.

Together, subdivision, bounds, topology and limit evaluation support ray–subdivision surface intersection: bounds eliminate irrelevant patches; surviving patches are refined, and regular candidate intersections are corrected using position and first derivatives. The returned intersection is a point of the limit subdivision surface with an error certificate, rather than automatically a point on the last control mesh.

Choose coefficients using the local subdivision matrix, not just by copying a pleasant-looking regular stencil. Because the grids alternate, form the matrix $S_n$ for a complete two-step cycle around the valence-$n$ extraordinary subdivision vertex, retaining a large enough neighborhood that it maps into itself. The three local coefficients enter this matrix through the specified vertex and face rules.

First require constant reproduction and commutation with affine maps: every target stencil must sum to one, and the valence-four rules must reduce to the regular ones already found. A propagation mask need not itself sum to one. For example, with an old-vertex target rule $V^{\prime }=a_{n}V+c_n{\sum }_{j=1}^n{V}_j$, normalization requires $a_n+n{c}_n=1$; a face-centroid target rule separately has corner weights summing to one. If face weights depend on incident valences, impose their sum on each actual face, rather than incorrectly using a propagation-column normalization. Rotational and reflection symmetry around the vertex should preserve an isotropic treatment of its sectors. Nonnegative coefficients are desirable for convex hull bounds and stability, although they are not by themselves a smoothness proof.

For convergence, seek a simple eigenvalue one corresponding to the constant eigenvector, with all remaining eigenvalues of modulus below one. For a well-defined tangent plane, seek two independent, semisimple tangent modes with a common positive subdominant eigenvalue:The angular modes should be the first sine and cosine modes around the ring. On the regular lattice the full two-step tangent scale is $1/2$, which provides a useful consistency check. These requirements suppress persistent unwanted modes and ensure that generic first-order geometry is dominated by a two-dimensional tangent space.

Then check that the characteristic map of a subdivision surface built from those tangent eigenvectors is regular and injective on the punctured neighborhood. A dominant tangent pair together with a regular, nonfolding characteristic map is the crucial $C^1$ target; eigenvalues alone are insufficient. This check rules out folds or vanishing tangent area that a spectrum cannot detect. For reasonable curvature behavior, also try to keep second-order modes at or below ${\lambda }^2$ in modulus, with equality and appropriate mode structure when nonzero finite curvature is desired. Avoid negative or complex modes that produce visible alternating ripples, excessive shrinkage and strongly anisotropic shape response.

These are selection criteria, not a uniquely determined triple of coefficients. The disappearance of extraordinary faces after one step simplifies the subsequent local matrix, but the extraordinary vertex remains and its smoothness still needs these tests. In particular, the regular $C^4$ result cannot simply be assigned to arbitrary valence.

A local subdivision matrix maps the control points in a refinement-invariant neighborhood to the refined neighborhood. Row sums one express constant reproduction. Its constant eigenvector has eigenvalue one; remaining eigenvalues govern decay of shape modes. Near an extraordinary subdivision vertex, tangent modes and a characteristic map of a subdivision surface are needed to test geometric smoothness, beyond simple convergence.