352 research outputs found
A matrix-based approach to properness and inversion problems for rational surfaces
We present a matrix-based algorithm for deciding if the parametrization of a
curve or a surface is invertible or not, and for computing the inverse of the
parametrization if it exists.Comment: 12 pages, latex, revised version accepted for publication in the
Journal AAEC
Reverse engineering of CAD models via clustering and approximate implicitization
In applications like computer aided design, geometric models are often
represented numerically as polynomial splines or NURBS, even when they
originate from primitive geometry. For purposes such as redesign and
isogeometric analysis, it is of interest to extract information about the
underlying geometry through reverse engineering. In this work we develop a
novel method to determine these primitive shapes by combining clustering
analysis with approximate implicitization. The proposed method is automatic and
can recover algebraic hypersurfaces of any degree in any dimension. In exact
arithmetic, the algorithm returns exact results. All the required parameters,
such as the implicit degree of the patches and the number of clusters of the
model, are inferred using numerical approaches in order to obtain an algorithm
that requires as little manual input as possible. The effectiveness, efficiency
and robustness of the method are shown both in a theoretical analysis and in
numerical examples implemented in Python
Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants
We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points.
Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring).
All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface.
For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system.
This yields an efficient, output-sensitive algorithm for
computing the discriminant polynomial
Implicitization of curves and (hyper)surfaces using predicted support
We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation.
For predicting the implicit support, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory.
Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial.
We implement our methods on Maple, and some on Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces.
We apply our approach to approximate implicitization,
and quantify the accuracy of the approximate output,
which turns out to be satisfactory on all tested examples; we also relate our measures to Hausdorff distance.
In building a square or rectangular matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complexes offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice.
We compare our prototype to existing software and find that it is rather competitive
Implicitization of surfaces via geometric tropicalization
In this paper we further develop the theory of geometric tropicalization due
to Hacking, Keel and Tevelev and we describe tropical methods for
implicitization of surfaces. More precisely, we enrich this theory with a
combinatorial formula for tropical multiplicities of regular points in
arbitrary dimension and we prove a conjecture of Sturmfels and Tevelev
regarding sufficient combinatorial conditions to compute tropical varieties via
geometric tropicalization. Using these two results, we extend previous work of
Sturmfels, Tevelev and Yu for tropical implicitization of generic surfaces, and
we provide methods for approaching the non-generic cases.Comment: 20 pages, 6 figures. Mayor reorganization and exposition improved.
The results on geometric tropicalization have been extended to any dimension.
In particular, Conjecture 2.8 is now Theorem 2.
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