Matrix representations by means of interpolation

Abstract: We examine implicit representations of parametric or point cloud models, based on interpolation matrices, which are not sensitive to base points. We show how interpolation matrices can be used for ray shooting of a parametric ray with a surface patch, including the case of high-multiplicity intersections. Most matrix operations are executed during pre-processing since they solely depend on the surface. For a given ray, the bottleneck is equation solving. Our Maple code handles bicubic patches in $\le 1$~sec, though numerical issues might arise. Our second contribution is to extend the method to parametric space curves and, generally, to codimension $> 1$, by computing the equations of (hyper)surfaces intersecting precisely at the given object. By means of Chow forms, we propose a new, practical, randomized algorithm that always produces correct output but possibly with a non-minimal number of surfaces. For space curves, we obtain 3 surfaces whose polynomials are of near-optimal degree; in this case, computation reduces to a Sylvester resultant. We illustrate our algorithm through a series of examples and compare our Maple prototype with other methods implemented in Maple i.e., Groebner basis and implicit matrix representations. Our Maple prototype is not faster but yields fewer equations and seems more robust than Maple's {\tt implicitize}; it is also comparable with the other methods for degrees up to 6.

• The paper introduces a robust interpolation matrix that circumvents the need for an explicit implicit equation by encoding geometric information in its kernel.
• It leverages sparse resultant theory and mixed symbolic-numeric computations to enable efficient ray shooting and manage high-multiplicity intersections.
• It extends the methodology to higher-codimension varieties using Chow forms and randomized algorithms to produce near-optimal implicit equations.

Matrix Representations of Geometric Objects via Interpolation

Overview

The paper "Matrix representations by means of interpolation" (1606.00789) advances the methodology of implicitization for parametric and point cloud geometric models. It constructs robust matrix representations that facilitate geometric operations, such as ray shooting and intersection computations, primarily through linear algebra and interpolation techniques. The approach is characterized by minimal sensitivity to base points and can rigorously handle cases with high-multiplicity intersections. Furthermore, the paper extends these techniques to higher-codimension varieties using Chow forms, proposing efficient randomized algorithms with guaranteed correct outputs.

Interpolation Matrices for Implicitization

The central contribution is an implicit matrix representation that circumvents the direct derivation of an explicit implicit equation for the geometric object. The constructed interpolation matrix is indexed by a monomial support derived from sparse resultant theory (for parametric cases) or by monomials up to a guessed total degree (for point cloud inputs). The kernel of the matrix encodes the coefficients of the implicit polynomial, even in the presence of base points. This aligns with the sparse elimination paradigm, where the sparsity of both parametric and implicit forms is exploited for computational tractability.

A salient aspect is the matrix's adaptation for both symbolic and numeric computation. All but one row are numeric, corresponding to evaluations of the parameterization (or samples of the point cloud), while the last row remains symbolic, accommodating substitution by an arbitrary geometric query (e.g., a ray).

Ray Shooting and Matrix Decomposition

The paper systematically addresses ray shooting at a (hyper)surface using only the interpolation matrix. The core idea is to encode the ray as a symbolic last row in the matrix and compute the determinant. Preprocessing using PLU or QR decomposition (on the numeric part) enables rapid determinant evaluation for arbitrary rays, reducing the online task to univariate polynomial solving.

Crucially, the method is agnostic to intersection multiplicity and resilient to base points — limitations of direct equation solving approaches. However, for large monomial supports, numerical instability can arise, particularly when using floating-point arithmetic in matrix decompositions. Inversion of the parameterization to verify the intersection's presence on a specific patch is acknowledged as a bottleneck, especially for high-degree surfaces.

Empirical evaluation using Maple demonstrates competitive performance for moderate problem sizes (e.g., bicubic patches), though state-of-the-art C++ implementations of syzygy-based matrix approaches outperform in both speed and numerical robustness.

Extension to Higher Codimension: Chow Forms and Randomized Construction

Addressing varieties of codimension greater than one, the paper leverages Chow forms — classically the foundation for the set-theoretic description of such varieties. The proposed algorithm constructs implicit (hyper)surface equations whose intersection recovers the parameterized variety. The approach is randomized: for space curves in $\mathbb{R}^3$, three conical surfaces are constructed via Sylvester resultant formulations, each corresponding to a random choice of apex outside the variety. Their intersection recovers the curve set-theoretically, with the method producing polynomials of degree close to optimal.

Notably, this methodology sidesteps the computationally intensive rewriting algorithms required in general Chow form expansion. Instead, it utilizes efficient resultant computations or interpolation guided by mixed volume bounds for the expected support. The algorithm consistently yields a near-minimal set of defining equations across tested examples, validated by experiments across dimensions.

Experimental Analysis and Complexity

Quantitative assessments reveal that for parameterizations of degree up to 6, the matrix interpolation method is competitive with methods based on Gröbner bases and syzygies in both runtime and output succinctness. For higher degrees, it produces a substantially lower number of implicit equations compared to native symbolic algebra systems, at the expense of increased per-equation computation time.

The method's asymptotic complexity is dominated by univariate solving (in ray shooting) and by resultant/interpolation computation in the higher-codimension case. Theoretical complexity matches standard expectations given polynomial degree and support size, with experimental runtimes corroborating practical feasibility up to moderately large examples (e.g., space curves of degree 18).

Theoretical and Practical Implications

The interpolation-matrix-centric framework provides:

• A unification of implicitization for parametric and point cloud data models under a sparse linear algebraic paradigm.
• Robustness to issues that typically challenge classical implicitization techniques, including base points and high-multiplicity intersections.
• A preprocessing-accelerated pipeline enabling efficient, repeated geometric queries (e.g., ray shooting without re-computation of the matrix core).
• Extension to higher-codimension varieties via practical Chow form constructions, thus enabling the description of complex curves and varieties with low-degree, minimal systems.

Future Developments

Open questions include tighter bounds on the degree and the number of implicit equations needed for arbitrary varieties, finer understanding and identification of extraneous factors in resultant/interpolation-based formulations, and exploration of hybrid symbolic-numeric computation to further scale the approach. The extension to intersection queries (e.g., surface-surface intersection with one surface given as a point cloud) evidences strong potential for broader application in computational geometric modeling.

Conclusion

The paper establishes a principled, efficient, and mathematically robust methodology for implicit matrix representations via interpolation. By integrating sparse elimination, interpolation, and Chow form theory, it enables efficient geometric operations on parametric and point-cloud-defined varieties, provides practical algorithms for higher-codimension cases, and serves as a solid foundation for further exploration in symbolic-numeric geometric computation.