Multi-Sided Quadratic Interpolants

• Multi-sided quadratic interpolants are a surface construction paradigm that creates smooth, vertex-interpolating surfaces from arbitrary meshes using quadratic patches.
• The framework employs partition-of-unity blending with local quadratic patches, integrating Bernstein bases and Wachspress coordinates to handle diverse mesh topologies.
• It efficiently manages non-uniform mesh valence and triangular subpatches, ensuring C² continuity and robust interpolation even on irregular polyhedral nets.

A multi-sided quadratic interpolant is a parametric surface construction that interpolates all vertices of a given closed mesh of arbitrary topology by blending locally defined, multi-sided quadratic patches. Salvi’s framework introduces a systematic method for generating $n$-sided local quadratic interpolants and assembling them via partition-of-unity techniques to obtain globally smooth, vertex-interpolating surfaces—even when mesh faces and one-ring neighborhoods are not restricted to quads. For non-quadrilateral domains, specialized parameterizations based on rational curves and Wachspress coordinates are employed, while subpatch treatments for triangular cases enable consistent handling of non-uniform mesh valence. The result is a flexible surface construction paradigm leveraging only quadratic local elements and smooth blending, with applicability to both regular and irregular polyhedral nets (Salvi, 27 Jan 2026).

1. Quadratic Local Interpolants on Regular Quadmeshes

In the context of a regular quadmesh (valence-4 at each vertex), a local quadratic interpolant is constructed as a tensor-product quadratic Bézier patch over a $3 \times 3$ control net. The mesh’s quad provides four corners $C_1, \ldots, C_4$, edge midpoints $E_1, \ldots, E_4$, and a face center $M$. Each $E_i$ is "lifted" such that the Bézier curve $[C_{i-1}, \hat{E}_i, C_i]$ interpolates the original $E_i$ precisely, with

$\hat{E}_i = 2E_i - \tfrac{1}{2}(C_{i-1} + C_i).$

The control net points $P_{ij}$ ($3 \times 3$0) are set in cyclic order, with the interior net point $3 \times 3$1 calculated as

$3 \times 3$2

Each local patch is defined over $3 \times 3$3 by the standard Bernstein basis:

$3 \times 3$4

where $3 \times 3$5, $3 \times 3$6, $3 \times 3$7.

To consistently blend corner-based interpolants around a quad, the domain is rotated for each local net, aligning evaluation points and ensuring that all nets interpolate common data at the correct parametric locations.

2. Blending Local Patches via Partition-of-Unity Schemes

The global surface patch on each quad is synthesized from four local corner interpolants mixed with a $3 \times 3$8 partition-of-unity "bump" (Hermite blend) function $3 \times 3$9:

$C_1, \ldots, C_4$0

commonly realized using a degree-$C_1, \ldots, C_4$1 Bernstein sum. The two-variable blend is $C_1, \ldots, C_4$2.

The blended surface then takes the form

$C_1, \ldots, C_4$3

where $C_1, \ldots, C_4$4 are rotated domain coordinates per corner. This construction guarantees $C_1, \ldots, C_4$5 continuity (typically $C_1, \ldots, C_4$6 with $C_1, \ldots, C_4$7) across quad boundaries, as both the function and $C_1, \ldots, C_4$8 derivatives vanish along patch seams by design.

3. Extension to General $C_1, \ldots, C_4$9-Sided Interpolants and Non-Regular Meshes

When mesh vertices possess valence $E_1, \ldots, E_4$0, the tensor-product construction does not generalize directly. The approach deploys a Quadratic Generalized Bézier (QGB) patch, interpolating the $E_1, \ldots, E_4$1-gon’s corners $E_1, \ldots, E_4$2, mid-edge points $E_1, \ldots, E_4$3, and center $E_1, \ldots, E_4$4 using Wachspress barycentric coordinates $E_1, \ldots, E_4$5 determined for the regular $E_1, \ldots, E_4$6-gon.

For side $E_1, \ldots, E_4$7,

$E_1, \ldots, E_4$8

describe, respectively, the position along the $E_1, \ldots, E_4$9th edge and the inward distance from that edge.

The QGB patch reads:

$M$0

where the central weight deficiency $M$1 ensures partition-of-unity and $M$2 is set as

$M$3

This formula specializes exactly to the tensor-product for $M$4 and to a standard quadratic Bézier triangle for $M$5.

A $M$6 rational mapping from $M$7 to kite-shaped subdomains of the $M$8-gon is essential to guarantee smooth glueing. This is achieved by constructing two "pencils" of rational quadratic Bézier curves indexed by $M$9 and $E_i$0, then locating their intersection to yield the mapped point in the polygon domain. While this step requires solving for rational curve intersections, the mapping is typically tabulated for efficiency.

4. Treatment of Triangular Subpatches and Alternatives

For $E_i$1, the QGB formula produces a quadratic Bézier triangle that lacks an interior control point and thus cannot interpolate the center $E_i$2. To resolve this, degree elevation to a cubic Bézier triangle introduces an interior control $E_i$3, yielding a patch with full cubic Bézier triangle evaluation using the triangle’s barycentric coordinates. The explicit formulas elevate the edge controls and calculate the unique interior as:

$E_i$4

Alternative subpatch schemes, such as "Midpoint" or "Midpoint-Coons" multi-sided patches, were considered. However, the QGB patch was found to be simpler to implement and slightly more efficient, with comparable surface quality empirically observed across various mesh typologies.

5. Algorithmic Workflow and Implementation Considerations

The step-by-step surface construction process over a triangle/quad mesh is as follows:

1. Convert any non-quad faces via a central split (equivalent to a single Catmull–Clark step), ensuring every face is a quad and preserving original vertices in the control net.
2. For each quad:
   • Gather one-ring data at each corner; for each, use tensor-product interpolation if valence is 4, or the corresponding QGB method otherwise.
3. Precompute the $E_i$5 rational mappings from $E_i$6 to kite-like subregions of the reference $E_i$7-gon for each valence, typically storing on a $E_i$8 grid for each valence to expedite inner-loop evaluation.
4. For each parameter $E_i$9, evaluate the four rotated local nets, blend by $[C_{i-1}, \hat{E}_i, C_i]$0, then sum to obtain $[C_{i-1}, \hat{E}_i, C_i]$1.
5. Stitch all patches. As blends vanish to second order at the edge, the resulting global surface is $[C_{i-1}, \hat{E}_i, C_i]$2 across patch interiors and $[C_{i-1}, \hat{E}_i, C_i]$3 at extraordinary (non-regular) points.

The dominant computational cost arises from four local quadratic patch evaluations, four blend evaluations, and, for irregular valencies, the resolution of two rational curve intersections per sample. In practice, the rational mapping lookups can be significantly accelerated with table-based approaches. All Bernstein-basis evaluations are numerically robust for degree $[C_{i-1}, \hat{E}_i, C_i]$4, and the continuity and support of blends can be adjusted by selecting the Hermite-blend order $[C_{i-1}, \hat{E}_i, C_i]$5. The methodology accommodates meshes with boundaries (via one-sided blends) and can incorporate additional constraints, such as normals or cross-derivatives, on mesh edges (Salvi, 27 Jan 2026).

6. Mathematical and Practical Context

The multi-sided quadratic interpolant paradigm significantly generalizes classical tensor-product Bézier patching to arbitrary-topology meshes while maintaining low patch degree and high surface quality. The methodology’s foundation in partition-of-unity blending, Wachspress coordinates, and rational geometric mappings allows for a direct, fully interpolating treatment of complex polyhedral networks. A plausible implication is that, due to the moderate degree and local nature of all interpolants, this approach maintains practical efficiency and robustness, with only modest computational overhead over traditional surface spline methods. The framework provides a bridge between established spline patch mechanisms and flexible, topology-agnostic geometric modeling, with particular relevance for applications requiring guaranteed interpolation and high smoothness of polyhedral input (Salvi, 27 Jan 2026).