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A geometry representation with scalar partition of unity basis functions commutes with every affine map of its control points. For $T(x)=Ax+b$, linearity moves $A$ through the sum and the sum-to-one identity supplies the translation $b$. Positivity is needed for a convex hull property, but not for this equivariance. The same argument applies to linear subdivision matrices whose rows sum to one. General projective transformations require homogeneous control points and rational dehomogenization rather than transforming Euclidean controls with unchanged weights.

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.

On a regular triangular grid, Loop subdivision retains an old vertex with weight $10/16$ and adds its six neighbors with weight $1/16$ each. A new edge vertex has weights $3/8$ on the endpoints and $1/8$ on the two opposite vertices. Extraordinary vertices require valence-dependent rules.

// Target: analysis.bigb

Finite local subdivision matrices for the scaled difference scheme track derivative variations in nested parameter intervals. A product-norm bound below one proves uniform contraction for every interval and therefore gives a lower bound on derivative continuity. A product eigenvalue of modulus at least one is an obstruction when its mode is reachable from the data and visible in the limit: generic data excite it, so the corresponding derivative differences fail to vanish. Harmless polynomial or unobservable modes must be removed before applying such an upper bound. Borderline smoothness needs direct analysis, not rounding a numerical exponent. Stability and convergence of the underlying scheme must be checked rather than inferred from mask width.

Fix an explicit stationary-grid convention. For subdivision arity $m>1$, write the univariate rule asA column of the subdivision matrix is one translated subdivision mask; the next column is translated by $m$ fine-grid rows. Thus the column displacement is the arity. If $a_k$ is active from $k=\alpha$ to $k=\beta$, the subdivision mask width is $w=\beta -\alpha +1$ fine-grid positions, after removing zero padding. Equivalently the mask span is $\beta -\alpha$ fine intervals. This is different from counting old control points in a row stencil: that count measures how many controls influence one newly created point. Stating the convention avoids confusing these two widths.

For a convergent subdivision curve scheme, its support is the influence region of one original control point, equivalently the support of a function of the basic limit $\varphi$ obtained from data $p_j={\delta }_{j0}$. Translation gives the representation $P(t)={\sum }_j{p}_j\varphi (t-j)$. Descendant indices of index zero have parameter positionsThus the limiting support enclosure is $[\alpha /(m-1),\beta /(m-1)]$, as in the support of a stationary subdivision scheme. Endpoint activity gives equality for the ordinary positive convergent masks under discussion.

The functional precision set of a subdivision scheme consists of functions exactly recovered from their uniformly sampled data using the scheme's consistent parameter placement. Polynomial precision records the exactly reproduced polynomial space. Sampling-grid spacing and origin must be specified; the usual arbitrary-grid precision requirement uses every spacing and shift, rather than merely membership in the fixed-grid reconstruction space.

Center the given mask at indices $-1,0,1$. Its binary linear interpolatory subdivision rule isEach new control point is either an old point or an edge midpoint. Hence every refined control polygon is the same curve with extra vertices. Starting from scalar impulse data, the limit is explicitlyThe mask width is three positions, its mask span is two, its largest row stencil uses two old control points, and its subdivision arity is two. Translating the mask indexing to $0,1,2$ translates the basic support to $[0,2]$; the support width remains two original grid intervals.

On $[j,j+1]$, reconstruction is $(j+1-t)p_j+(t-j)p_{j+1}$. Constant and affine samples are therefore recovered exactly. A non-affine polynomial cannot agree with a linear expression on a whole open interval. Thus its polynomial functional precision set of a subdivision scheme isFor continuous functions reproduced at every translated sampling grid, the same conclusion holds: agreement with all chord interpolants forces the affine interpolation identity, hence affinity. At one fixed grid, the larger exactly representable class is the space of linear splines with those spline knots; this is not a higher polynomial precision claim.

For arbitrary data, the one-sided derivatives at $j$ are $p_j-p_{j-1}$ and $p_{j+1}-p_j$. They need not coincide. Therefore the generic limit is $C^0$, not $C^1$. Its first derivative is piecewise constant; exceptional affine data, or individual matching neighboring slopes, can be smoother. This conclusion follows directly from the limit formula, not just from a heuristic mask-width test.

A solid body transform is a rigid transformation: it preserves distances within the object. In three-dimensional coordinates it has the form $T(x)=Rx+b$, with translation $b\in {\mathbb{R}}^3$ and an orthogonal matrix $R$. For a proper physical rotation, the coefficient constraints areEquivalently the columns of $R$ are an orthonormal basis with positive orientation. A reflection also preserves distance but has $\det R=-1$ and is excluded when “solid-body motion” means a rotation and translation. There are three rotational and three translational degrees of freedom, not twelve independent matrix coefficients.

Computationally use the homogeneous coordinates representationThe last row is $(0,0,0,1)$; matrix multiplication composes transforms. Alternatively store a unit quaternion together with $b$, maintaining its unit norm. A transpose gives the rotational inverse: $T^{-1}(y)=R^T(y-b)$.

Write the cubic Bézier curve as $P(t)={\sum }_{i=0}^3{B}_i^3(t)p_i$, where $B_i^3(t)=\left(\genfrac{}{}{0ex}{}{3}{i}\right)(1-t{)}^{3-i}{t}^i$. The binomial theorem gives ${\sum }_i{B}_i^3(t)=1$. Transforming the controls producesThis proves pointwise equivariance, including the original parameter values; it is stronger than a statement that the two curves merely have the same shape.

Orthogonality was not used in the calculation. The identical proof applies to every affine map $T(x)=Ax+b$, including scales, shears and even singular affine maps. It applies to every representation $P(u)={\sum }_i{\varphi }_i(u)p_i$ with scalar, control-independent partition of unity basis functions: Bézier curves of every degree, B-splines, surfaces built on a tensor-product surface basis, triangular Bézier patches, and more general linear control-point representations. Positivity is not needed for affine equivariance of a geometric basis. For linear stationary subdivision matrices with row sums one, each refinement step commutes with the affine transformation; taking a convergent limit proves the corresponding assertion for subdivision curves and subdivision surfaces. Rules with geometry-dependent nonlinear weights require separate equivariance tests and are not covered automatically.

The affine class is the largest class that preserves every such Euclidean-control construction with unchanged basis weights, under the usual continuity assumption. In fact the degree-one case would forcefor all $x,y$ and $0\le t\le 1$. This segment identity, followed by continuity, forces $T$ to be affine. Nonlinear transforms such as perspective division generally fail it.

There is a further extension when the representation itself is enlarged: apply a projective transformation linearly to homogeneous controls $(w_i{p}_i,w_i)$, then dehomogenize the weighted sum. The transformed objects are rational Bézier curves or non-uniform rational B-splines, with transformed weights as well as transformed Euclidean controls. This works wherever the new denominator is nonzero. It does not assert projective invariance of ordinary polynomial Bézier curves with their old weights.

An efficient subdivision surface interrogation interface needs the mesh topology and patch adjacency, boundary and trimming information, patch-local parameter charts and their transition maps, and local restriction/refinement with the complete influencing control net. Numerical enquiries should supply limit positions $S$, first derivatives $S_u,S_v$, and second derivatives where defined; these determine normal vectors and curvatures at regular points. It must also return certified patch bounding volumes and convergent geometric/derivative error bounds. Near an extraordinary subdivision vertex, use the local subdivision matrix and limit/tangent evaluation appropriate to that scheme; second derivatives need not exist merely because a tangent plane does. Finite refined mesh vertices are not exact limit positions.

Represent the cutting plane as $n\cdot x=d$, with $n$ normalized. On each chart setThe desired plane section of a subdivision surface is the image of the zero set $g=0$. For a bounding volume $B$, compute a certified interval $[g_{\min },g_{\max }]$ for the plane functional over $B$. If that interval is strictly positive or strictly negative, reject the patch. With positive partition-of-unity weights, the minimum and maximum of $n\cdot p_i-d$ over all influencing control points already bound its limit patch. For masks without this convex hull property, the interface must provide a different valid enclosure.

An adaptive algorithm that isolates components before tracing is:

This is exhaustive to a chosen tolerance when certified subdivision isolates finitely many regular transverse branches and enclosures converge. If $\Vert \nabla g\Vert \ge \gamma >0$ locally and $\Vert DS\Vert \le M$, a residual tolerance $|g|\le \delta$ corresponds locally to a geometric correction bounded by $M\delta /\gamma$, using the gradient flow toward the level set. Bounds on second derivatives similarly control curve-segment chord error bound. As $\gamma$ approaches zero, a fixed residual tolerance no longer gives a fixed geometric error, so the step and stopping rules must reflect conditioning.

Seeding must not rely just on corner signs. For example $S(u,v)=(u,v,u^2+v^2-r^2)$ cut by $z=0$ has a closed circle wholly inside a patch whose corner values are all positive. Certified range tests plus subdivision retain it; a corner-only marching rule can miss it. Tangencies also change the nature of the output: $S=(u,v,u^2+v^2)$ meets $z=0$ in one isolated point, while a coplanar patch contributes an area rather than a curve. Thus a general plane section is not necessarily a union of regular curves; the algorithm should classify these cases instead of silently fabricating or deleting a branch.

For efficiency, cache subdivision stencils, patch extraction, limit-evaluation coefficients and bounds; reuse child data instead of reconstructing the entire refined mesh. Use local influencing neighborhoods, with the halo dictated by mask support, and keep adjacency/boundary topology separate from changing geometric coordinates. Plane-functional bounds can be computed by refining scalar signed-distance controls, which is cheaper than refining all three coordinates. Combine coarse inexpensive boxes with tighter local convex hull or interval arithmetic tests only where needed. Adapt steps to curvature and transversality, reuse branch seeds, and use regular spline evaluation away from extraordinary points. Stable edge ownership, outward-rounded bounds, scale-aware tolerances, and explicit degeneracy flags are correctness requirements as well as performance issues. The result is a limit-surface intersection computation; merely intersecting a finitely refined control polyhedron with the plane supplies only an uncontrolled approximation.

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.

There are no numerical continuity bounds determined by $w$ and $a$ alone: the subdivision mask coefficients matter. First check constant reproduction, ${\sum }_j{M}_{ϵ-aj}=1$ for every residue $ϵ$, and convergence. For the usual stationary scalar scheme, two useful calculations are a uniform contraction test and a spectral obstruction test. Higher smoothness is tested with derived difference subdivision masks.

Write $M(z)={\sum }_j{M}_j{z}^j$ and $s_a(z)=1+z+\cdots +z^{a-1}$. Using backward differences $\Delta P_i=P_i-P_{i-1}$, the generating-function identityproves the subdivision difference scheme relation $\Delta S_M=S_{M/s_a}\Delta$. Iterating gives ${\Delta }^r{S}_M=S_{M/s_a^r}{\Delta }^r$. To test a candidate $C^r$ limit, suppose the requisite sum rules hold, so $M/s_a^{r+1}$ is a finite Laurent polynomial. The scaled derivative controls $q_i^{\ell }=a^{r\ell }{\Delta }^r{P}_i^{\ell }$ are refined by the subdivision maskwhose residue sums are one. Their first differences are refined byThe powers of $a$ are essential because each refinement shrinks parameter intervals by $a$.

A lower continuity bound comes from proving uniform contraction. The infinity norm of one difference-refinement step is bounded byIf $q<1$, adjacent derivative controls contract geometrically at every location. More generally, form the $p$-step subdivision maskand test ${\max }_{ϵ\mathrm{mod}{a}^p}{\sum }_j|(B_r^{[p]}{)}_{ϵ-a^{p}j}|<1$. This proves convergence of the constant-preserving derivative scheme to a continuous limit. One justification is to compare consecutive piecewise-linear interpolants: their difference is bounded by a constant times the maximum adjacent-control difference, and the geometrically decreasing bound is summable. To identify the limit as $P^{(r)}$, use fixed-degree Cardinal B-spline interpolants of the original refined controls. Their $r$th derivatives are sums of the scaled finite differences; convergence of both functions and derivatives gives $P\in C^r$. Testing successive $r$ supplies a guaranteed lower bound.

An upper continuity bound comes from an observable noncontracting shape mode. Build finite local subdivision matrices for $B_r$ on a refinement-invariant control neighborhood. For a digit $ϵ$ their entries are $(A_{ϵ}{)}_{ij}=(B_r{)}_{ϵ+i-aj}$, since the new index is $ak+ϵ+i$. Restrict to the realizable difference space and discard modes that do not affect the limit. A word ${ϵ}_1,\dots ,{ϵ}_p$ corresponds to a nested parameter location. If its product has an observable eigenvalue $\lambda$ with $|\lambda |>1$, repetition of this word makes some scaled derivative differences grow rather than tend to zero. For a stable limit representation, $C^r$ regularity would force these differences to vanish; hence general-position data cannot have $C^r$ continuity. A modulus-one mode requires examination of whether it actually persists, rather than an automatic rounding rule.

The two calculations can be organized through the joint spectral radius $\hat{\rho }$ of these restricted matrices:These norm and spectral bounds for subdivision regularity make the test computable from the subdivision mask. The right bound certifies all-word contraction when it is less than one; the left bound detects a repeated-word obstruction when it exceeds one. Longer products improve the tests. A single matrix's spectrum is not a substitute for controlling arbitrary products. The product estimates also appear in Charina's analysis of subdivision regularity; the strict contraction and observable-mode arguments above explain their respective roles.

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.

A subdivision surface is the limit of repeated mesh refinement and vertex averaging. Exact interrogation evaluates this limit and, where defined, its tangent vectors and normal vectors; a finite refined control mesh is generally only an approximation to the limit.

// Target: analysis.bigb

Interrogation of a subdivision limit needs patch/topology navigation, refinement, limit evaluation and derivatives where they exist, with certified bounds and geometric error estimates. On regular patches one may use the corresponding spline representation. At extraordinary subdivision vertices, local subdivision matrices determine limit positions and tangents under the scheme's smoothness conditions. A normal vector may exist even when second derivatives and Gaussian curvature do not; finite refined polygons are approximations rather than exact limit queries.