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The radial basis function interpolation is an effective method for approximation on scattered data. However, this interpolation method suffers from the contradiction between the accuracy and the numerical stability. With considering the distribution of the interpolation points, a bounded random shape variation scheme for the radial basis function is developed to circumvent the problem of numerical stability. In this scheme, the shape variation is bounded by the value that is determined by the maximum distance and the minimal distance which are applied to describe the average density of the interpolation centers. Within this bound, the shape of the MQ is modified through a random scheme. With applying this bounded randomly variable shape scheme, the accuracy and the stability of the MQ interpolation are balanced. Comparisons on the accuracy and the condition number of the interpolation matrix between the constant shaped MQ interpolation and this bounded randomly variable shaped MQ interpolation have been made to verify the conclusion. Furthermore, this scheme is integrated in the dual reciprocity boundary element method in the analysis of three dimensional elastic problems. Results of the numerical examples demonstrated that the developed scheme improved the accuracy of the dual reciprocity boundary element method stably.

In this note, we extend work of Farkas and Rim��nyi on applying quadric rank loci to finding divisors of small slope on the moduli space of curves by instead considering all divisorial conditions on the hypersurfaces of a fixed degree containing a projective curve. This gives rise to a large family of virtual divisors on $\overline{\mathcal{M}_g}$. We determine explicitly which of these divisors are candidate counterexamples to the Slope Conjecture. The potential counterexamples exist on $\overline{\mathcal{M}_g}$, where the set of possible values of $g\in \{1,\ldots,N\}$ has density $��(\log(N)^{-0.087})$ for $N>>0$. Furthermore, no divisorial condition defined using hypersurfaces of degree greater than 2 give counterexamples to the Slope Conjecture, and every divisor in our family has slope at least $6+\frac{8}{g+1}$., Revised according to referee comments, to appear in Experimental Mathematics

In this paper, a new refined quasi-3-dimensional logarithmic shear deformation theory (RQ-3DLSDT) and an advanced moving Kriging interpolation (AMKI) based-meshless method is combined for the first time to study the static bending, free vibration, and compressive buckling analysis of isotropic and sandwich functionally graded plates laid on elastic foundations. The RQ-3DLSDT considering the effect of thickness stretching based only on four-unknown variables to determine the displacement field leading to reduce computational efforts. The weak forms of the equilibrium equation are discretized and numerically solved by the AMKI mesh-free method employed a proposed quadric correlation function for getting stable solutions.

Orthoptic curves for the conics are well known. It is the Monge's circle for ellipse and hyperbola, and for parabola it is its directrix. These conics are level sets of quadratic functions in the plane. We consider level sets of quadratic functions in higher dimension, known as quadric hypersurfaces. For these hypersurfaces we present and study their orthoptic sets, which extend the idea of orthoptic curves for conics.

This paper is concerned with the problem of multi-view 3D reconstruction with an un-calibrated micro-lens array based light field camera. To acquire 3D Euclidean reconstruction, existing approaches commonly apply the calibration with a checkerboard and motion estimation from static scenes in two steps. Self-calibration is the process of simultaneously estimating intrinsic and extrinsic parameters directly from un-calibrated light fields without the help of a checkerboard. While the self-calibration technique for conventional (pinhole) camera is well understood, how to extend it to light field camera remains a challenging task. This is primarily due to the ultra-small baseline of the light field camera. We propose an effective self-calibration method for a light field camera for automatic metric reconstruction without a laborious pre-calibration process. In contrast to conventional self-calibration, we show how such a self-calibration method can be made numerically stable, by exploiting the regularity and measurement redundancies unique for the light field camera. The proposed method is built upon the derivation of a novel ray-space homography constraint (RSHC) using Plucker parameterization as well as a ray-space infinity homography (RSIH). We also propose a new concept of “rays of the absolute conic (RAC)” defined as a special quadric in 5D projective space $${\mathbb {P}}^5$$ . A set of new equations are established and solved for self-calibration and 3D metric reconstruction specifically designed for a light field camera . We validate the efficacy of the proposed method on both synthetic and real light fields, and have obtained superior results in both accuracy and robustness.

A Minkowski plane is Euclidean if and only if at least one ellipse is a quadric. We discuss the higher dimensional consequences too.

Signal modeling is an important technique in many engineering applications. This paper is concerned about signal modeling problem for the sine multi-frequency signals or periodic signals. In terms of different characteristics between the signal output and the signal parameters, a separable modeling scheme is presented for estimating the signal parameters. In order to seize the real-time information of the signals to be modeled, a sliding measurement window is designed for using the observations dynamically and implementing accurate parameter estimates. Because the amplitude parameters are linear with respect to the signal output and the angular frequency parameters are nonlinear with respect to the signal output, the signal parameters are separated into a linear parameter set and a nonlinear parameter set. Based on these separable parameter sets, a nonlinear optimization problem is converted into a combination of the optimization quadric and the nonlinear optimization. Then, a separable multi-innovation Newton iterative signal modeling method is derived and implemented to estimate sine multi-frequency signals and periodic signals. The simulation results are found to be effective of modeling dynamic signals. For the reason that the proposed method is based on dynamic sliding measurement window, it can be used for online estimation applications.

Let $$X\subset \mathbb {P}^3$$ X ⊂ P 3 be an integral and non-degenerate curve. We say that $$q\in \mathbb {P}^3\setminus X$$ q ∈ P 3 \ X has X-rank 3 if there is no line $$L\subset \mathbb {P}^3$$ L ⊂ P 3 such that $$q\in L$$ q ∈ L and $$\#(L\cap X)\ge 2$$ # ( L ∩ X ) ≥ 2 . We prove that for all hyperelliptic curves of genus $$g\ge 5$$ g ≥ 5 there is a degree $$g+3$$ g + 3 embedding $$X\subset \mathbb {P}^3$$ X ⊂ P 3 with exactly $$2g+2$$ 2 g + 2 points with X-rank 3 and another embedding without points with X-rank 3 but with exactly $$2g+2$$ 2 g + 2 points $$q\in \mathbb {P}^3$$ q ∈ P 3 such that there is a unique pair of points of X spanning a line containing q. We also prove the non-existence of points of X-rank 3 for general curves of bidegree (a, b) in a smooth quadric (except in known exceptional cases) and we give lower bounds for the number of pairs of points of X spanning a line containing a fixed $$q\in \mathbb {P}^3\setminus X$$ q ∈ P 3 \ X . For all integers $$g\ge 5$$ g ≥ 5 , $$x\ge 0$$ x ≥ 0 we prove the existence of a nodal hyperelliptic curve X with geometric genus g, exactly x nodes, $$\deg (X) = x+g+3$$ deg ( X ) = x + g + 3 and having at least $$x+2g+2$$ x + 2 g + 2 points of X-rank 3.

We investigate necessary conditions for Gorenstein projective varieties to admit semiorthogonal decompositions introduced by Kawamata, with main emphasis on threefolds with isolated compound $A_n$ singularities. We introduce obstructions coming from Algebraic $\mathrm{K}$-theory and translate them into the concept of maximal nonfactoriality. Using these obstructions we show that many classes of nodal threefolds do not admit Kawamata type semiorthogonal decompositions. These include nodal hypersurfaces and double solids, with the exception of a nodal quadric, and del Pezzo threefolds of degrees $1 \le d \le 4$ with maximal class group rank. We also investigate when does a blow up of a smooth threefold in a singular curve admit a Kawamata type semiorthogonal decomposition and we give a complete answer to this question when the curve is nodal and has only rational components., Final version, to appear in Moscow Mathematical Journal

The problem of classifying linear systems of conics in projective planes dates back at least to Jordan, who classified pencils (one-dimensional systems) of conics over $${\mathbb {C}}$$ C and $$\mathbb {R}$$ R in 1906–1907. The analogous problem for finite fields $$\mathbb {F}_q$$ F q with q odd was solved by Dickson in 1908. In 1914, Wilson attempted to classify nets (two-dimensional systems) of conics over finite fields of odd characteristic, but his classification was incomplete and contained some inaccuracies. In a recent article, we completed Wilson’s classification (for q odd) of nets of rank one, namely those containing a repeated line. The aim of the present paper is to introduce and calculate certain combinatorial invariants of these nets, which we expect will be of use in various applications. Our approach is geometric in the sense that we view a net of rank one as a plane in $$\mathrm {PG}(5,q)$$ PG ( 5 , q ) , q odd, that meets the quadric Veronesean in at least one point; two such nets are then equivalent if and only if the corresponding planes belong to the same orbit under the induced action of $$\mathrm {PGL}(3,q)$$ PGL ( 3 , q ) viewed as a subgroup of $$\mathrm {PGL}(6,q)$$ PGL ( 6 , q ) . Since q is odd, the orbits of lines in $$\mathrm {PG}(5,q)$$ PG ( 5 , q ) under this action correspond to the aforementioned pencils of conics in $$\mathrm {PG}(2,q)$$ PG ( 2 , q ) . The main contribution of this paper is to determine the line-orbit distribution of a plane $$\pi $$ π corresponding to a net of rank one, namely, the number of lines in $$\pi $$ π belonging to each line orbit. It turns out that this list of invariants completely determines the orbit of $$\pi $$ π , and we will use this fact in forthcoming work to develop an efficient algorithm for calculating the orbit of a given net of rank one. As a more immediate application, we also determine the stabilisers of nets of rank one in $$\mathrm {PGL}(3,q)$$ PGL ( 3 , q ) , and hence the orbit sizes.

The μ-basis is a developing algebraic tool to study the expressions of rational curves and surfaces. It can play a bridge role between the parametric forms and implicit forms and show some advantages in implicitization, inversion formulas and singularity computation. However, it is difficult and there are few works to compute the μ-basis from an implicit form. In this paper, we derive the explicit forms of μ-basis for implicit monoid curves and surfaces, including the conics and quadrics which are particular cases of these entities. Additionally, we also provide the explicit form of μ-basis for monoid curves and surfaces defined by any rational parametrization (not necessarily in standard proper form). Our technique is simply based on the linear coordinate transformation and standard forms of these curves and surfaces. As a practical application in numerical situation, if an exact multiple point can not be computed, we can consider the problem of computing “approximate μ-basis” as well as the error estimation., Agencia Estatal de Investigación

[Abstract] We analyze the characteristic polynomial associated to an ellipsoid and another quadric in the context of the contact detection problem. We obtain a necessary and sufficient condition for an efficient method to detect contact. This condition, named smallness condition, is a feature on the size and the shape of the quadrics and can be checked directly from their parameters. Under this hypothesis, contact can be noticed by means of the expressions in a discriminant system of the characteristic polynomial. Furthermore, relative positions can be classified through the sign of the coefficients of this polynomial. As an application of these results, a method to detect contact between a small ellipsoid and a combination of quadrics is given.

Boundary representation (B-rep) model editing plays an essential role in computer-aided design and motivates the very recent direct modeling paradigm, which features intuitive push-pull manipulation of the model geometry. In mechanical design, a substantial part of B-rep models being used are quadric models (composed of linear and quadric surfaces). However, push-pulling such models is not trivial due to the possible smooth face-face connections in the models. The major issue is that, during push-pull moves, it is often desirable to preserve these connections for functional, manufacturing, or aesthetic reasons, but this could cause complex inconsistencies between the geometry and topology in the model and lead to robustness issues in updating the model. The challenge lies in effectiveness towards detecting the instants when geometry-topology inconsistencies occur during push-pull moves. This paper proposes a novel reverse detection method to solve the challenge and then, based on it, presents a robust method for push-pull direct modeling while preserving smooth connections. Case studies and comparisons have been conducted to demonstrate the effectiveness of the method., 20 pages

Real-analytic CR functions on real-analytic CR singular submanifolds are not in general restrictions of holomorphic functions, unlike in the CR nonsingular case. We give a simple condition that completely characterizes those quadric CR singular manifolds of codimension 2 in ${\mathbb C}^{n+1}$ for which an extension result holds. Consequently, we obtain an extension result for general real-analytic CR singular submanifolds of codimension 2. As applications we give a condition for the flattening of such submanifolds, and we classify CR singular images of CR submanifolds up to second order., Comment: 22 pages, accepted to Math. Z, fix hypothesis on Proposition 7.1

We give a solution to the weak Schottky problem for genus five Jacobians with a vanishing theta null, answering a question of Grushevsky and Salvati Manni. More precisely, we show that if a principally polarized abelian variety of dimension five has a vanishing theta null with a quadric tangent cone of rank at most three, then it is in the Jacobian locus, up to extra irreducible components. We employ a degeneration argument, together with a study of the ramification loci for the Gauss map of a theta divisor., Comment: 18 pages, comments very welcome

Let f: C --> P^3 be a general curve of genus g, mapped to P^3 via a general linear series of degree d; and let Q be a general (and thus smooth) quadric. In this paper, we show that the points of intersection f(C) \cap Q give a general collection of 2d points on Q, except for exactly six exceptional cases. We also prove similar theorems for every other pair (r, n) for which, except for only finitely many pairs (d, g), the intersection of a general curve of genus g mapped to P^r via a general linear series of degree d, with a general hypersurface S of degree n, is a general collection of dn points on S. As explained in arXiv:1809.05980, these results play a key role in the author's proof of the Maximal Rank Conjecture

In this paper, we present a new connection between representation theory of noncommutative hypersurfaces and combinatorics. Let $S$ be a graded ($\pm 1$)-skew polynomial algebra in $n$ variables of degree $1$ and $f =x_1^2 + \cdots +x_n^2 \in S$. We prove that the stable category $\mathsf{\underline{CM}}^{\mathbb Z}(S/(f))$ of graded maximal Cohen--Macaulay module over $S/(f)$ can be completely computed using the four graphical operations. As a consequence, $\mathsf{\underline{CM}}^{\mathbb Z}(S/(f))$ is equivalent to the derived category $\mathsf{D^b}(\operatorname{\mathsf{mod}} k^{2^r})$, and this $r$ is obtained as the nullity of a certain matrix over ${\mathbb F}_2$. Using the properties of Stanley--Reisner ideals, we also show that the number of irreducible components of the point scheme of $S$ that are isomorphic to ${\mathbb P}^1$ is less than or equal to $\binom{r+1}{2}$., Comment: 10 pages, v2: minor changes

We consider the problem of calculating a refractive element with two surfaces, forming a flat front and a given distribution of illumination. The supporting quadrics method is formulated for calculating a given optical element and it is shown that this method coincides with the gradient method for some functional related to the problem of the Monge-Kantorovich mass transfer problem. This enables adaptive selection of the step in the supporting quadric method. At the end of the article a design example is given.

In this paper we remove the solid incidence assumption in the characterization of J. Schillewaert (A characterization of quadrics by intersection numbers, Des. Codes Cryptogr., 47 (2008), 165–175) by proving that quadric plane incidence numbers imply quadric solid incidence numbers, except for the dual complete 11-cap of PG(4, 3). Furthermore, new characterizations of the parabolic quadric Q(4, q) and the ovoidal cone of PG(4, q) are provided.

Images of a nearby celestial body collected by a camera on an exploration spacecraft contain a wealth of actionable information. This work considers how the apparent location of the observed body's horizon in a digital image may be used to infer the relative position, attitude, or both. When the celestial body is a sphere, spheroid, or ellipsoid (as is the case for most large bodies in the Solar System), the projected horizon in an image is a conic-usually an ellipse at large distances and a hyperbola at small distances. This work develops non-iterative and analytically exact methods for every case (all combinations of unknown state parameters and quadric shapes), completely superseding older horizon-based methods that are iterative, approximate, or both. Some of the analytic methods presented in this work are new. Recognizing that these developments build on techniques that may be unfamiliar to many spacecraft navigators, this work is fashioned as a tutorial. Descriptive illustrations and numerical examples are provided to make concepts clear and to validate the proposed algorithms.

We present a simple yet efficient algorithm for recognizing simple quadric primitives (plane, sphere, cylinder, cone) from triangular meshes. Our approach is an improved version of a previous hierarchical clustering algorithm, which performs pairwise clustering of triangle patches from bottom to top. The key contributions of our approach include a strategy for priority and fidelity consideration of the detected primitives, and a scheme for boundary smoothness between adjacent clusters. Experimental results demonstrate that the proposed method produces qualitatively and quantitatively better results than representative state-of-the-art methods on a wide range of test data.

This paper considers a feedback control law that achieves attitude stabilization for Earth-pointing spacecraft using only magnetorquers as torque actuators. The control law is proportional derivative (PD)-like with matrix gains, and it guarantees asymptotic stability. The PD matrix gains are determined through the numerical solution of a periodic linear quadric regulator problem. A case study shows the effectiveness of the considered control law, and specifically of the gain selection method, in a simplified simulation scenario.

Given a degree one del Pezzo surface with canonical singularities, the linear series generated by twice the anti-canonical divisor exhibits the surface as the double cover of the quadric cone branched along a sextic curve. It is natural to ask if this description extends to the boundary of a compactification of the moduli space of degree one del Pezzo surfaces. The goal of this paper is to show that this is indeed the case. In particular, we give an explicit classification of the boundary of the moduli space of anti-canonically polarized broken del Pezzo surfaces of degree one as double covers of degenerations of the quadric cone., Comment: 10 pages; comments welcome

Let [math] be a number field or a [math] -adic field and [math] the function field of a curve over [math] . Let [math] be a prime. Suppose that [math] contains a primitive [math] -th root of unity. If [math] and [math] is a number field, then assume that [math] is totally imaginary. In this article we show that every element in [math] is a symbol. This leads to the finite generation of the Chow group of zero-cycles on a quadric fibration of a curve over a totally imaginary number field.

We present a novel and effective method for detecting 3D primitives in cluttered, unorganized point clouds, without axillary segmentation or type specification. We consider the quadric surfaces for encapsulating the basic building blocks of our environments - planes, spheres, ellipsoids, cones or cylinders, in a unified fashion. Moreover, quadrics allow us to model higher degree of freedom shapes, such as hyperboloids or paraboloids that could be used in non-rigid settings. We begin by contributing two novel quadric fits targeting 3D point sets that are endowed with tangent space information. Based upon the idea of aligning the quadric gradients with the surface normals, our first formulation is exact and requires as low as four oriented points. The second fit approximates the first, and reduces the computational effort. We theoretically analyze these fits with rigor, and give algebraic and geometric arguments. Next, by re-parameterizing the solution, we devise a new local Hough voting scheme on the null-space coefficients that is combined with RANSAC, reducing the complexity from $O(N^4)$ to $O(N^3)$ (three points). To the best of our knowledge, this is the first method capable of performing a generic cross-type multi-object primitive detection in difficult scenes without segmentation. Our extensive qualitative and quantitative results show that our method is efficient and flexible, as well as being accurate., Submitted to IEEE Transactions on Pattern Analysis and Machine Intelligence (T-PAMI). arXiv admin note: substantial text overlap with arXiv:1803.07191

In this paper, we study real hypersurfaces in the complex quadric space $$Q^m$$ whose structure Jacobi operator commutes with their structure tensor field. When the normal vector field is $$\mathfrak {A}$$ -principal we show that the Reeb curvature $$\alpha $$ is non-vanishing and determine principal curvatures of the hypersurface. In the case of $$\mathfrak {A}$$ -isotropic normal vector field, we prove that the hypersurface is Hopf if it has vanishing Reeb curvature or commuting shape operator. We also consider Reeb flat hypersurfaces, namely when the Reeb curvature is zero. We see that this family of hypersurfaces is non-empty and among other results we prove that if the Ricci tensor of a Reeb flat Hopf hypersurfaces is Killing, then the Ricci tensor is parallel.

Tracking the trajectory of a moving body is one of key issues for a multiple freedom motion joint with multi-point surface contacts. In general, the tangent space of the motion can be determined by...

Motivated by understanding the quality of tractable convex relaxations of intractable polytopes, Ko et al. gave a closed-form expression for the volume of a standard relaxation Q ( G ) of the Boolean quadric polytope (also known as the (full) correlation polytope) P ( G ) of the complete graph G = K n . We extend this work to structured sparse graphs. In particular, we (i) demonstrate the existence of an efficient algorithm for vol ( Q ( G ) ) when G has bounded treewidth, (ii) give closed-form expressions (and asymptotic behaviors) for vol ( Q ( G ) ) for all stars, paths, and cycles, and (iii) give a closed-form expression for vol ( P ( G ) ) for all cycles. Further, we demonstrate that when G is a cycle, the simple relaxation Q ( G ) is a very close model for the much more complicated P ( G ) . Additionally, we give some computational results demonstrating that this behavior of the cycle seems to extend to more complicated graphs. Finally, we speculate on the possibility of extending some of our results to cactii or even series–parallel graphs.

We compute the Chow-Witt rings of split quadrics over a field of characteristic not two. We even determine the full bigraded I-cohomology and Milnor-Witt cohomology rings, including twists by line bundles. The results on I-cohomology corroborate the general philosophy that I-cohomology is an algebro-geometric version of singular cohomology of real varieties: our explicit calculations confirm that the I-cohomology ring of a split quadric over the reals is isomorphic to the singular cohomology ring of the space of its real points., Comment: This is the final version. Some additional details have been added, and a sign ambiguity of the previous version has been resolved

A quadratic set of a nonsingular quadric Q of Witt index at least three is defined as a set of points intersecting each subspace of Q in a possibly reducible quadric of that subspace. By using the theory of pseudo-embeddings and pseudo-hyperplanes, we show that if Q is one of the quadrics $$Q^+(5,2)$$ , Q(6, 2), $$Q^-(7,2)$$ , then the quadratic sets of Q are precisely the intersections of Q with the quadrics of the ambient projective space of Q. In order to achieve this goal, we will determine the universal pseudo-embedding of the geometry of the points and planes of Q.

We consider a family $M_{t}^{3}$, with $t>1$, of real hypersurfaces in a complex affine three-dimensional quadric arising in connection with the classification of homogeneous compact simply connected real-analytic hypersurfaces in $\mathbb{C}^{n}$ due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the Cauchy–Riemann (CR)-embeddability of $M_{t}^{3}$ in $\mathbb{C}^{3}$. In our earlier article, we showed that $M_{t}^{3}$ is CR-embeddable in $\mathbb{C}^{3}$ for all $1. In the present paper, we prove that $M_{t}^{3}$ can be immersed in $\mathbb{C}^{3}$ for every $t>1$ by means of a polynomial map. In addition, one of the immersions that we construct helps simplify the proof of the above CR-embeddability theorem and extend it to the larger parameter range $1.